Meta-Learning Theory-Informed Inductive Biases using Deep Kernel Gaussian Processes
Authors: Bahti Zakirov
Gasper Tkacik
Venue: ICLR 2026 Poster (2026)
Keywords: Computational Neuroscience
Gaussian Processes
Efficient Coding
Deep Kernel Learning
Meta-Learning
Inductive Biases
Bayesian Deep Learning
Primary area: applications to neuroscience & cognitive science

Abstract: Normative and task-driven theories offer powerful top-down explanations for biological systems, yet the goals of quantitatively arbitrating between competing theories, and utilizing them as inductive biases to improve data-driven fits of real biological datasets are prohibitively laborious, and often impossible. To this end, we introduce a Bayesian meta-learning framework designed to automatically convert raw functional predictions from normative theories into tractable probabilistic models.
We employ adaptive deep kernel Gaussian processes, meta-learning a kernel on synthetic data generated from a normative theory. This Theory-Informed Kernel specifies a probabilistic model representing the predictions of a given theoretical model: usable for both fitting data and rigorously validating the theory. As a demonstration, we apply our framework to the early visual system, using efficient coding as our normative theory. 
We show improved response prediction accuracy in ex vivo recordings of mouse retinal ganglion cells stimulated by natural scenes compared to conventional data-driven baselines, while providing accurate uncertainty estimates and interpretable representations. Using exact Bayesian model selection, we also show that our informed kernel can accurately infer the degree of theory-match from data, confirming faithful encapsulation of theory structure. This work provides a more general, scalable, and automated approach for integrating theoretical knowledge into data-driven scientific inquiry in neuroscience and beyond.

Published as a conference paper at ICLR 2026
META- THEORY-INFORMED INDUCTIVE BIASES USING DEEP KERNEL GAUSSIAN PROCESSES
Bahti Zakirov
Gaäper Tkačik
Institute of Science and Technology Austria bzaki rovê! at..ac.at
Institute of Science and Technology Austria gtkac k at.
ABSTRACT
Normative and task-driven theories offer powerful top-down explanations for biological systems, yet the goals of quantitatively arbitrating between competing theories, and utilizing them inductive biases improve datven fits of real biological datasets are prohibitively laborious, and often impossible. To this end, we introduce Bayesian meta-learning framework designed to automatically convert raw functional predictions from normative theories into tractable probabilistic models. We employ adaptive deep kernel Gaussian processes, meta-learning a kernel on synthetic data generated from a normative theory. This Theory-Informet Kernel specifies probabilistic model representing the predictions of given theoretical model: usable for both fitting data and rigorously validating the theory. As demonstration, we apply our framework to the early visual system, using efficient coding our normative theory. We show improved response prediction accuracy in ex vivo recordings of mouse retinal ganglion cells stimulated by natural scenes compared to conventional data-driven baselines, while providing accurate uncertainty estimates and interpretable representations. Using exact Bayesian model selection, also show that our informed kernel can accurately infer the degree of theory-mated from data, confirming faithful encapsulation of theory structure. This work provides more general, scalable, and automated approach for integrating theoretical knowledge into data-driven scientific inquiry in neuroscience and beyond.
Black Box Theory
Theory-Generater Meta Train Set
Probabilistic Model Representing Theory
i) Quantify Theory Match to Real Data
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Figure 1: Overview of the Proposed Framework.
INTRODUCTION
striking feature of living matter is teleonomy, or apparent goal-directedness. This idea permeates quantitative life sciences as a predictive principle: by assuming organisms evolve under selection for functional phenotypes, we can model their structure and behaviours as optimized solutions to ecologically relevant tasks (Bialek 2012). This longstanding tradition, often called normative, or more recently task-driven modeling, has deep roots in computational neuroscience and sensory coding theory (Barlow et al. 1961 Atick 1992), having provided top-down explanation for the structure and organization of receptive fields in early visual areas such as the retina and primary visual cortex (Olshausen & Field 1996 Karklin & Simoncelli 201T). It has also proven directly

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predictive of activity in higher visual, somatosensory. and auditory areas often outperforming purely data-driven models (Yamins et al. 2014 Kell et al. 2018; Zhuang et al. 2021 Vargas et al. 2224))
Despite mounting evidence demonstrating the broad predictive and explanatory power of top-down models, this enterprise suffers from two important challenges. First, given multiple competing normative theories of the same system, there is general, principled procedure in neuroscience to quantitatively arbitrate between them. Solutions like Bayesian Model Selection (Kass & Raftery 1995) do exist, but are challenging to apply due to their requirement of first hand-specifying probabilistic model representing the theory knowledge, and the typical intractability of the resulting marginal likelihoods, which are difficult even estimate. Second, despite calls in scientific machine learning for domain-informee' inductive biases, it remains difficult to systematically combine the theoretical knowledge of good top-down model with the noisy, complex reality of empirical data to aid data-driven predictive inference. Current approaches to solving these challenges require laborious expert hand-design, and are either somewhat heuristic and gystem-speciffe (for example, constraining
the receptive fields of models of early visual systems (Qiu et al., 2023b Goldin et al. 2023 or
limited in applicability to idealized simple models (MHynarski et al. 2021).
Here, we identify Bayesian meta-learning as a principled methodological solution to both problems. We present a framework designed to automatically construct theory- infory-informed probabilistic models applicable to any system for which top-down theories that can generate input/output predictions are available. We demonstrate our framework using a bespoke variant of adaptive deep kernel meta- learning (Patacchiola et al.) 2020 Chen et al. (023): meta-learning a Gaussian process kernel from synthetic data generated by the well-known efficient coding normative theory of the retina. Concretely, we first meta-learn deep feature extractor shared across synthetic tasks derived from efficient coding theory. By virtue of meta-learning, this feature extractor forms an abstract metric embedding where the geometric distances between transformed inputs are meaningful for the entire class of functions consistent with our instantiation of the theory. The frozen embedding is subsequently used and updated by downstream task-adaptive components to form a AnteoryIInfomel Kernel for biological data. Our informed kernel defines a Gaussian process prior over the space of functions, or an inductive bias, representing the thcorized biological purpose of the system.
We show that this theory-informed inductive bias outperforms conventional data-driven baselines for learning the neural encoding function of ex vivo retinal ganglion cells, providing evidence that our framework fruitfully transfers knowledge from the theoretical model to real biological data. Further, the probabilistic theory-informed models produced by our framework natively allow the exact computation of the Marginal I. Likelihoods needed for Bayesian model comparison, thereby enabling rigorous information-theoret quantification of the match between theoretical model and real data. We show that this is sensitive enough to accurately infer the degree of match between theory and data, which is more challenging than a simple binary choice between competing models. All in all, our work provides much needed bridge between top-down theory-driven understanding, and bottom-up data-driver prediction (Breiman 2001
Our main contribution is framework enabling new scientific capabilities for neuroscience that is built upon and contributes to recent advances in Bayesian machine learning. For neuroscience, we introduce a path to automatically construct tractable probabilistic models from "black-box" theories (Figs. enabling the rigorous, information-theorre (Fig. in a way that is insensitive to the thcory's parameterization. Our theory-informn improves predictive accuracy and uncertainty quantification on real data (Fig. 3) while also providing interpretability and allowing the embedded theory knowledge to be automatically relaxed or enriched provide compelling new
validation of competing scientific hypotheses
model
as the data demands (Fig. 4). For Bayesian machine learning. application of deep kernels for the open problem of domain inffaind kernel design. We demonstrate successful real-world implementation: introducing a bespoke architecture and other practical design choices useful for future work (S.F.4) that avoid the common "feature collapse" pathology (Ober) ct al.) (202I) to deliver improved uncertainty quantification (Fig. Bp.c) Finally, we validate key Bayesian deep learning principles (Wilson & Izmailov] 2020) relating generalization to inductive biases (Fig. (3) on a challenging scientific problem.
2 BACKGROUND
Top-Down Theories in Neuroscience. Normative theories (also known as optimality theories)
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propose that the properties of biological systems emerge result of evolutionary pressures optimizing a utility functional. In these frameworks, the parameters governing neural coding are regarded as optimal solutions to the underlying ecological tasks. Formally, one may express this as 0" = arg max9 U [P(r|or 0), Pa (a)] where r is the neural response, I: denotes the stimulus drawn from a naturalistic distribution (x)))), 0 are the model parameters, and U a carefully designed utility functional that quantifies how well the response meets the system's theorized ecological or behavioral goals. Of particular relevance is the development of efficient coding theories in early
sensory systems (Barlow al. [961] Atick 1992.
Olshausen & Field 1996; Karklin & Simoncelli
2011; Karklin & Lewicki, 2009| Attneave 1954 Simoncelli & Olshausen 2001; Lewicki [002]]
Hyvärinen al. 2009). This line of research was among the first derive the structure of retinal and primary visual cortical encoding by assuming that it optimizes information transmission using limited neural activity. Contemporary task-driven extensions operationalize these normative principles by optimizing flexible models, often deep neural networks, directly on ecologically relevant task performance, proving highly predictive of neural activity across various domains (Yamins et al. 2014
Kell et al. 2018; Vargas et al. 2024).
Gaussian Processes (GPs). GPs define distributions over functions such that any finite set of points
drawn from the process has joint Gaussian distribution. In machine learning. GPs serve as Bayesian priors for functions/ I in regression and classification models (Williams Rasmussen [006]] MacKay 2003). Concretely, we place GP prior fe (O0 Ko (:--)) with desitive
kernel Kepp, and link fo to observed data y through a Gaussian likelihood for regression:
y(x)) =N(fz,o,,
To compute the final predictive equations of all GP's in this work, GP priors
are conditioned on training data D Purain = {X. y) via Bayesian inference: P(y.|X., Durai)) K(X,, X.) K(X,, (X(, X) +
N (K(X,, X) (XX, X) + o
where "=" denotes "test data", and ((),,X2) is a matrix shaped |X]] x [x2]. The kernel entirely specifies the inductive bias, since it determines the mean and covariance functions of the underlying Gaussian prior. Deep Kernels improve GP expressivity by parameterizing the kernel with a neural network (Wilson et al. 2016). Hyperparameters 8gp of the kernel and likelihood can be selected by maximizing the marginal likelihood of the data:
P 0) = / Plulfip(fl0004
Meta-Learning, The goal of meta-learning is to use labeled input/output data from tasks in a meta- meta-train set T -= =[ y/ ) NN (where Ny denotes the number of tasks, and n; the number of data points per task) to learn inductive biases that are useful for fitting on other tasks in the meta-test set (Caruana 1997 Zhang & Yang 2018). Individual tasks are split into support and query data (i.e. n, n n where the support set is analogous to the training data for that task, and query to the validation data. Meta-learned models may contain task-adaptive components, which (unlike fixed meta-learned components) can be re-fit every individual task, often on different optimization objectives.
THFORYDD META-LEARNING FRAMEWORK
The framework's 's application has three steps (Fig. To: generating synthetic data from (normative) theory, meta-learning a theory-informed kernel from the synthetic data, and adapting/applying the resulting GP prior to real data.
Our Infore-Informed Kernel comprises disjoint meta-learned and task-adaptive modules (Fig. 2
A meta-learned high-capacity feature extractor d, shared across tasks, that learns an abstract metric embedding where the geometric distances between inputs are meaningful for the entire class of theory-consistent functions
A task-adaptive linear head (per task). PN that adapts the initial metric embedding. Here, the (hr } serve two important purposes. First, by enabling task-specific adaptations of the metric embedding they help mitigate negative transfer (Standley et al.) (020)) alleviating the scenario where different tasks have slightly incongruent structure, thereby requiring customized embeddings. Second, the additional flexibility helps adapt the structure learned
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Adsk-Ad Deep Kernel Meta-Learning Approach. Inputs (e.g.. images) pass through shared feature extractor whose weights feeta are meta-learned across tasks. The resulting features are passed to linear head h; and ultimately the Gaussian process (parameters gg..) layer, both of which are adapted separately to every task for which we generate predictions (e.g., neural activity). Together, these components constitute the Theory-Informed Kernel.
Figure 2:
entirely synthetic data to the idiosynerasies of real biological measurements- thereby bridging the gap between simulated and real data.
A task-adaptive Gaussian process layer (per task) that makes predictions by conditioning the theory-informed prior on the transformed inputs. Here we use the Radial Basis Functions ker- NNr -==21²²) with hyperparameters 6gp = {(a o,)) 1
nel KRBF(2, z'; ap)) = of exp
22
(denoting the output scale, length scale, and likelihood noise level respectively).
The meta-training procedure (Algorithm [ employs bi-level optimization akin (Chen al. 2023): first an inner loop fits the task-specific linear head (h,) and GP hyperparameters (Ugp.) using support data for each task by optimizing Eq. [T] then an outer loop updates the shared feature extractor (ø) based on the adapted model's 's log likeliooo on query data. Once trained, is frozen, while task-adaptive modules freshly adapt the prior to any meta-test task (S.F. Eq D. The final Theory-Informed Kernel (TIK) specifying the prior for each task i is thus
KTIK,(Xx,2)) = oz exp
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EXAMPLE: CHEORY-INFORMEE KERNEL FOR EFFICIENT CODING IN THE RETINA
We demonstrate our framework on mouse retinal ganglion cell (RGC) responses to natural images.
4.1 BIOLOGICAL DATA AND PREPROCESSING
We gratefully use previously published vivo calcium imaging data from 86 RGCs of single mouse retina responding to 50-frame natural movie sequences presented at 30Hz in UV channels (Qiu et al. 2023a). Inputs are derived from downsampled 36 x 32 pixel movie frames. A single measured fluorescence response (output E RR is recorded per 50 preceding movie frames for each neuron, as proxy for that neuron's 's activity (S.D). Each neuron's responses all stimuli constitute separate task, and we use the terms neurons and fasks interchangeably throughout this section. The dataset came pre-split (16200/750 train/test+val) and prepreprocessed. To adapt the data for image-based models, we temporally flattened the movies using a canonical temporal filter derived from Linear-Nonlines (LN) model fits to the neural data S.D.2.. We then eliminated near duplicaee images via hierarchical clustering (S.D3),, yielding (1452/ (400,350) train/(test, val)) unique images x8 We therefore have 86 biological meta-test tasks. each corresponding to the responses (E R) of one biological neuron 1452 natural images R 36x22, From machine-learning perspective, this corresponds to
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Figure 3: Theory-Informed Kernel improves fit on retinal data. (a) Test accuracy of model predictions at varying train set sizes N (mean Pearson correlation over 86 neurons, standard error over 5 seeds). (b) Posterior mean (solid line), epistemic uncertainty (shaded), posterior samples (colored lines) of tested GP models over the first principal component (PC1) of train images. Models conditioned on N = 60 (top) and N = 1400 (bottom) points (red dots). Projected full training images shown as gray ticks (above x-axes). (c) Kernel Density Estimate (KDE) of the NLPD scores; lower is better) of the RBF and the Theory-Informed kernels (all 86 neurons, 5 seeds).
86 scalar-output regression tasks, each with 1452 input-output pairs {Iin ) where In E IR 3R:3x32 is the nith preprocessed image and Vi,r E IR is the measured response to image n of neuron i.
4.2 CONSTRUCTING THE THEORY-INFORMES META-TRAIN SET
To synthesize the meta-train set, we adapted the efficient coding (EC) normative model of Ocko et al. (Ocko et al.) 2018). This model is a convolutional autoencoder trained on natural images with the FC objective of maximizing reconstruction accuracy while minimizing activity in the autoencoder bottleneck layer given constraints. A single synthetic meta-train task, T;, corresponds learning the linearized encoding function of single optimized bottleneck neuron. Given set of N natural images (I', 36 32 pixels), each task is to predict the N corresponding scalar responses, defined by the dot product V f 2g., where is the neuron's specific receptive field (RF). The linearization isolates the well-validated (Hyvärinen et al.] 2009; Karklin & Simoncelli] (011)) center-surroue
RFs that emerge the single-neure consequences of the population-lever efficint coding objective. We extracted these RFs (ft) from the optimized bottleneck by fitting an LN model to its activations. We then applied standard data augmentation and preprocessing extend the meta-train set, resulting in final 490 synthetic meta-train tasks derived from EC theory. Full details of this procedure, including rationale and sensitivity analysis, are provided in S.E
4.3 THEORY-INFORMET KERNEL IMPROVES LEARNING
Following Sec. 3 meta-learn the feature extractor (ő) on our theory-generated data (Sec. 4.2
Freezing Φ. we fit the task-adaptive components (h; and gg..) for each neuron i (Nys== 86) using [ To assess the transfer of our meta-learned inductive bias, we then compare performance metries against standard neuroscience baselines. Our first baseline is the well-validated Systems Identification convolutional neural network (CNN), specifically designed for this task (Qiu al., 2023b). Our second baseline is the standard Radial Basis Functions (RBF) GP, with kernel parameters set by optimizing Eq. [ We also compare against the Linear-Nonling
(LN) model, transfer learning, and ablated versions of our own model, designed to ensure that the transferred features reflect knowledge obtained from the theory (S.B). We quantified the test accuracy using the Pearson correlation between predicted and measured neural responses. Hyperparameters were identified via cross validation; carly stopping was used for our validation data (350 samples), equally for all models (S.G.2).
Fig. 3a depicts the performance of our model alongside the baselines, as a function of the number of training data points N. The experimental results follow clear trends, with the RBF GP performing competitively with the CNN at low data sizes (N < 64). and the LN outperforming the Theory- Informed Kernel in extremely low data regimes, where it is difficult to learn even the linear heads (N 8). Nonetheless, across large range in N, we find that model outperforms all baselines
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(Fig. 3.. S.BB). Because our approach ultimately feeds the data embeddings obtained by the meta- learned feature extractor & and linear heads h; into its own RBF GP, the improved performance over the bare RBF GP operating directly on the image imputs indicates that the learned embedding is relevant to modelling the structure in the biological data. Because the T'heory-Inooma significantly outperforms ablated versions where either randomize or remove the meta-learned (S.B) and the meta-train set, we conclude that Φ indeed encapsulates knowledge from our instance of efficient coding theory, thereby improving performance by higher margin than learning the task task-adaptive parameters alone, or simply meta-learning on generic task would allow.
Kernel
Comparatively, our model achieves both high data efficiency for small N, as well as the highest accuracy using the large N (full dataset). This supports the contemporary narrative in probabilistic deep learning: that well-calibrated but flexible inductive biases are key to powerful models that perform well with any amount of data (Wilson & Izmailov 20200.
4.4 THEORY-INFORMED KERNEL PRESERVES EPISTEMIC UNCERTAINTY
A crucial property of probabilistic methods like ours is their ability to represent uncertainty in their predictions. In uncertainty quantification (UQ), an important distinction is made between epistemic and aleatoric uncertainties (Hüllermeier & Waegeman 202I]. with the former being unpredictability reducible by observing more data or improving model specification, and the latter being irreducible unpredictability (c.g.. noise). Due to pathologies that arise from the interaction between deep learning and the marginal likelihood objective (Eq. I Deep Kernel GPs notoriously tend to exhibit feature collapse, specifically resulting in overconfident epistemic uncertainty (Van Amersfoort al. 202T Liu et al. 2020 Ober et al. 2021). Naturally, this is a concern for our framework.
To assess whether compare it to the RBF GP, with both models fit on the full train set. We first present, on the best fit neuron for both models (i 37), visualizations of their predictions and 95% CI (here only epistemic uncertainty), as function of the first principal component (PC1) of the full stimulus train set (fixing the other PCs to their means). We show cases where both models are conditioned on either or N = 1400 data (Fig. 3b). Both models exhibit the characteristic reduction of uncertainty near the observed data at N 60, but unlike the RBF GP, our preserves some uncertainty near PC1 = 0 where the samples are dense. Conditioned on N = 1400 data, the epistemic uncertainty of the RBF GP collapses near the data, while the informed kernel maintains reasonable confidence region enveloping substantial proportion of the data (despite only showing the epistemic part). This suggests, qualitatively, our representation improves uncertainty representation, reducing overconfidence in regions with many observations.
model's representation supports reasonable uncertainty quantification, we
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Informed Kernel heteroskedastically
Next, to quantitatively assess UQ in our model, we compute the standard Negative Log Predictive Density (NLPD) metric on the full images: NLPD [model,)
log P(ttesta Furain, iraii..; gg.g..i hs) For neuron i, P(ytes.sl-) denotes the GP posterior predictive distribution obtained by conditioning GP; on Dirain. Evaluating this posterior at M test inputs yields an Mddimensional multivariate Gaussian predictive distribution A(n, E,), and the NLPD is the negative log-robbablityy of the observed test responses under this Gaussian. Lower NLPD indicates more probability mass on the true responses, reflecting both mean accuracy and uncertainty calibration. For this metric, we consider both the epistemic and alleatoric uncertain- ties. Fig. 3 shows the kernel density estimate of NLPD values across all neurons and data. Our Theory- Kernel exceeds the UQ capabilities of the RBF baseline on biological data, avoiding common Deep Kernel GP pitfalls (Oher et al., (2021) and enabling potential downstream applications where it is vital for the model to "know what it doesn't know" (Cowley et al. 2017 Frazier 2018).
4.5 THEORY-INFORMET KERNEL LEARNS INTERPRETABLA REPRESENTATIONS
How exactly does our model's learned representation support its predictive performance? As a Deep Kernel GP with RBF layer, its predictions are driven by the pairwise Euclidean distances between transformed inputs. To gain mechanistic insight into how the task-specific linear transformations h, tailor the model's function, we analyze their impact on these critical pairwise distances. Starting with a set of probe images म, we compute a prototype image, Pt, for each h; by summing the pixel-level overlap masks between pairs of images Im, In E weighted by how h, alters their

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Cete with Oread Filerr
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Figure 4: Task- Task-Adaptin Adaptive Heads Learn Interpretable Representations. (a) Prototype images (P;) derived from learned representations ground truth linear filters of synthetic neurons. (b) Mean Pearson coir. between the absolute values the P. (blue) and filters learned by an LN model (black) with ground truth in (a), varying train size N. (c) P: for the 25 best (dark blue) and worst (light blue) performing real neurons, sorted by NLPD, learned from the full dataset (N = 1452). (d) Mean Pearson corr. between P; at varying N, with the P; full dataset (N 1452), split according to NLPD as in (c). (b) and (d) averages over neurons with standard errors across 5 random seeds.
These P. represent features of the inputs that h; learn to 'pull together' or 'push apart' for their task i, in their output representation distances.
projected distances ():())))) - (.(xx)))
S.H.I).
We first inspect the prototype images on synthetic data, finding that prototype images P. recapitulate the filters used to generate the theory-informed responses very well (Fig. 4a). We further observe that the rate which the prototype image of the Theory-Informan the ground truth RF filter as function of train set size N closely matches the rate of recovery by the LN model's s linear filter (Fig. D)) indicating the importance of P; this model. At high train set size N, however, the correlation between the ground truth and prototype image stops increasing monotonically, in a textbook example the Bayesian Occam effect (Rasmussen Ghahramani 20000; MacKay 1992 2003). As N grows, the GP layer's non-parametric capacity automatically expands, so the marginal likelihood stops rewarding additional structure in h; and favors less informed prior. This contextualizes
Kernel recovers (in absolute value)
argument from Sec. |4.3] about informed yet flexible models that retain their advantage across all dataset sizes: when data are scarce, the theory-informer inductive bias supplies the needed structure; when data plentiful, Eq. B aatomatically relaxes that structure permit predictive fit.
Applied biological data, we find clear distinction between P; corresponding tasks on which the model performed well, and those on which it performed poorly. The 25 best performing (by NLPD) h; transform the data in interpretable ways: resembling what may be the receptive fields of the biological neurons (Fig. 4c). The 25 worst h, conversely, seemingly fail to do so. This is reflected in (Fig.) 4i), which depicts the Pearson correlation between the prototype images learned at various N, with the prototype image given the full data; the worst performing tasks learn a less interpretable and also less consistent prototype image. Unlike the synthetic case, real neurons the still in prior-aided regime
structure in P; keeps increasing up N 1024, indicating that where the data alone have not saturated the non-pant non-parametric capacity.
Taken together, our Theory-Informee Kernll framework is both performant and interpretable. When fails, we gain insight into why; when it succeeds, we can watch the Occam effect unfold, demon- strating that Bayesian principles deliver practical gains in real- world settings. Moreover, prototype- basset based interpretation is readily transferable to domains beyond sensory neuroscience, wher- ever one wishes to visualize what deep-kernel GP has learned to look for.
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Figure Theory Informed Kernel quantifies Theory Fidelity. (a) Degree of optimality, 3 inferred by our framework, plotted against ground truth synthetic neurons (black dots; Pearson correlation = 0.88). Subpanel: example synthetic receptive fields of varying optimality, (b) Scree plot with the inferred efficient coding level of optimality 8 for all 86 recorded retinal ganglion cells. Inset: "Prototype images" P top three RGCs (detailed in Fig. 4. Right margin: KDE of inferred optimality for all 86 RGCs (blue) and 100 synthetic optimal neurons (green).
4.6 THEORY-INFORMED KERNEL ENABLES VALIDATION OF BLACK BOX NORMATIVE THEORIES
major methodological challenge in top-down (either normative or task-driven) modelling the validation of how well hypothesized optimality theory describes the biological system. Here, we cast the problem of quantifying the degree to which an optimality theory describes the real dataset as Bayesian Model Comparison (BMC). Typically. BMC involves comparing the marginal likelihood, P(D)M), which quantifies the evidence for competing hypotheses (models M) given the data Bayesian might decide between M 1 (c.g., specific theory) and M = 0 (c.g., null hypothesis) by selecting the highest scoring M. In our GP framework, this suggests a comparison between the theory-informan kernel and strong yet theory-agnostic null model kernel like the RBF. However, simple binary choice between these two alternatives may inadequate for biological systems, which often only partially adhere to the postulated theoretical principles (e.g., neurons sometimes exhibit deviations from the poooretically-rrddictt center-surround receptive fields). We thus propose Kg,, interpolating between theory-informed (TIK) and theory-agnostic (RBF) kernels:
K Ka( x') 3KTi((; x') + (1 30-)3 KRBF(Z,,')
Since the marginal likelihood quantifies the evidence for the specific prior defined by K3, the optimal 3* arg max; P(Y indicates the degree which the data provides evidence for the theory- informed structure beyond what can be captured by the powerful-yet-generi, null kernel. In other words, 8 be interpreted the inferred measure of the theory's relevance, alternatively, the system's degree of postulated optimality.
We first validate this approach on synthetic tasks. We generate these by starting with 20 optimal synthetic receptive fields (RFs) unseen during meta-training, and gradually add structured noise to push them away from optimality (S.144). As proxy for the ground truth optimality of these new synthetic RFs, we use the standard procedure of computing the R² of a fit of the Difference-of- Gaussians (DoG) model to RF, as DoG are theorized to constitute the efficient coding solution for the retina (S..4) Atick & Redlich 1992). We then freeze the feature extractor & and fit the task- adaptive parameters (linear heads h, and GP parameters OGP,)) ) for KYIK, as well as parameters gmull of KRBF. separately on each task Finally, with all parameters frozen for both kernels, we find optimal 3 = arg 10 X,, 3) via grid search over 100 points. Fig. 5h shows that 8*, inferred neuron by neuron as described, correlates strongly with the ground truth optimality level on synthetic data. Given postulated normative theory, 8 therefore approximates the neuron's degree of optimality. We validate the importance of theory-informed features with an ablation study S.I
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Next, we repeat the same procedure on our biological RGC dataset, by estimating S for each neuron 1. The inferred optimality levels of all 86 neurons are presented in Fig. [5p, as an example of a scientifically relevant amalysis enabled by our Bayesian framework. Most neurons score highly on the proposed optimality metric 3", consistent with the general consensus about the applicability of the efficient coding theory (Simoncelli & Olshausen 200T; Pitkow & Meister 2012). For comparison,
Fig. 55p also depicts, in the right margin, the 3 of 100 randomly selected synthetic optimal neurons (unseen in meta-training).
The ability of our framework to quantify how well a top-down theory explains neural data is a practical step forward for neuroscience. Though BMC has long been advocated as the principled solution to this problem (Jaynes] 2003), its routine use has been hindered by two obstacles: (i) manually translating a normative theory into a predictive probabilistic model; and (ii) computing the resulting marginal likelihoods in high-dimensional neural datasets. Consequently, most studies still rely on heuristic comparisons of summary statistics, such as the receptive field structure or tuning curve parameters (Ringach et al. 2002, Karklin & Simoncelli 2011). By embedding normative theories into a Gaussian process framework and leveraging the exact tractability of Gaussian marginal likelihoods, our approach addresses both obstacles. This makes principled Bayesian model comparison more feasible at scale and lowers the barrier for quantitatively testing normative theories on neural data.
DISCUSSION AND CONCLUSION
Related Work. Concepts of meta-learning and domain-informer inductive biases pervade science, and scientific machine learning (Kariiaaakis et al. [02T Lake et al. 2017 McCoy et al. 2020). Our framework was inspired by Linderman & Gershman (2017)ss call to embed high-level computational goals of neural systems into statistical models. In line with their Bayesian tenets, where theory (via priors) is combined with data (via likelihoods), our work delivers on their promises: enhancing interpretability and identifiability, and enabling empirical validation of theories. We also build on the elegant framework of Mlynarski et al. (202I), who incorporate maximum entropy priors derived from top-down theories into parametric Bayesian models, thereby improving inference and validating optimality theories using BMC. Similarly, Bittner et al. (2021) use normalizing flows to represent the distribution of mechanistic model parameters supporting desired emergent computational properties. In contrast, our approach sidesteps the need to commit to any specific parametric form, by meta- learning the priors directly in function space, yielding increased generality and scalability without compromising Bayesian principles. We are indebted to the work of Qiu et al. (2023b). who use efficient coding as an inductive bias for predictive modelling on their RGC dataset which we use here as well. Their approach hinges on enforeing shared spatial filters for two CNN models trained in parallel: one performing efficient coding (EC), the other fitting data, yielding improved accuracy and biologically plausible CNN filters. Unlike Qiu et al, who specifically design an inductive bias to exploit EC's ability to produce useful spatial filters for CNN, our framework does not assume the biological system to be modeled or the theory knowledge to be meta-learned, thus promising applicability to a broader class of systems. Finally, our GP architecture is variant of Chen et al.) (2023) meta-learner, to which we have introduced the ht (crucial to bridge the sim-to-real gap), and adjusted the training to avoid feature collapse (..F4.. Architecture aside, our work fundamentally differs: focusing on theory to data transfer rather than data-driven meta-learning. We elaborate in (S.A), particularly contrasting our work with that of Qiu and Miynarski.
Weaknesses and Future Directions. (i) A way to inspect what meta-learned knowledge is actually stored in the shared feature extractor & would be very useful, especially for domains other than efficient coding where predictive aspects of normative theories are less well understood. Although this a generie limitation of meta-learnin, the inherent interpretability of deep kernel representations, demonstrated in Sec. 4.5) may provide a stepping stone for future work. (ii) In seeking a principled separation between meta-learned knowledge and task-specific adaptations, we deliberately used frozen, meta-learned feature extractor ø in our kernel. An exciting open question is whether the knowledge in & could be enriched by carefully allowing its adaptation also on biological data. (iii) Predictions relating to Bayesian Model Comparison suffer from all of the commonly known challenges of using B.MC with powerful models (Kass & Raftery 1995 Gelman et al.) 1995 [Bernardo] & Smith 2009): namely, sensitivity to data properties, to the null model, and to the adaptation of the prior. Finally, (iv), though our framework is by design general, our empirical validation is limited to single challenging system. We justify our claim to generality in (..C)

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While there many prospective improvements that are interesting research directions, view them as promising future extensions to our framework and not as refutations of its contributions.
Conclusion. We have presented Bayesian framework to meta-learn flexible probabilistic models from top-down theories. Using deep kernel GPs, we' ve successfully transferred knowledge from an efficient coding theory of early visual processing model vivo neural recordings, demonstrating interpretability and improving accuracy, uncertainty quantification, and data efficiency compared baselines and ablations. Through Bayesian model comparison, our framework provides new scalable path to quantitatively validate normative theories against empirical data-an advancement with far-reaching potential for neuroscience.
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REPRODUCIBILITY STATEMENT
All resources required for the reproduction of our results are detailed within this paper and its supplementary material. The biological data, sourced from Qiu et al. (2023b a). is publicly available; our specific preprocessing pipeline, including temporal filtering and image selection, is described in The synthetic meta-training dataset was generated using an efficient coding autoencoder, with the full procedure, model architecture, and data augmentation detailed in S.E. Our primary model's architecture, meta-training algorithm (Algorithm(]) and task-adaptation procedure
S.D
specified in For baselines, complete implementation details and hyperparameter settings, derived from systematic grid searches, provided in S.G.2 A comprehensive summary of all hyperparameters used across all experiments is available Table S6 Our codebase can be found here) with the latest URL maintained in the Xiv version.
S
ETHICS STATEMENT
This research exclusively utilized a pre-existing, publicly available dataset of ex vivo mouse retinal ganglion cell recordings (Qiu et al.) a)) No new animal experiments were conducted for this study. The original data collection was performed by the cited authors under their institution's approved protocols for animal welfare. Our work is therefore confined to the computational analysis of this existing data. Beyond the use of this dataset, our research does not present foreseeable ethical concerns.
5.1 ACKNOWLEDGEMENTS
This research was funded in whole in part by the Austrian Science Fund (FWF) 10.55776/OE16. We thank Wiktor Młynrski and Henry Moss for helpful discussions. We thank Max Cairny-Leeming and James Odgers for proofreading the manuscript.
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SUPPLEMENTARY MATERIAL: META LEARNING THEORY INFORMED INDUCTIVE BIASES USING DEEP KERNEL GAUSSIAN PROCESSES
S.A
Extended Related Work
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S.B
Extended Figure 3a and Ablation Study Results
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S.B.1
Comparison to Transfer Learning
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S.B.2
Analysis of relative sources of inaccuracy of different models
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S.C
On the Generality of our framework
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S.C.1
Deep Kernels define broadly applicable function space priors
S.C.2
Generality via Modular Meta-Learning Pipeline
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S.D
Biological Data Preprocessing
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S.D.1
Dataset Source and Initial Processing by Qiu et al.
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S.D.2
Temporal Filter Extraction via LN Model
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Filtering non redundant images
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S.E
Synthetic Meta-Train Dataset Generation
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S.E.1
Normative Model (Efficient Coding Autoencoder)
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S.E.2
Data Augmentations Performed to extend the Meta- Train set
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Meta-Learmn Algorithm and Training Details
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Model Architecture
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Meta- Training Procedure
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Task Adaptation Procedure
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Design Choices Mitigating Feature Collapse in our Deep Kernel model
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Implementation Details and Runtimes
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Software and Hardware
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Baseline and Ablation Model Details
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Runtimes
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Prototype Extraction and Interpretability Analysis
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Computation Prototype Images
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Alternative Normalizations for Computing Prototypes
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Additional Visualizations
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Bayesian Model Comparison and Optimality Validation
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Ablation Study on Bayesian Model Comparison
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Mixture Kernel Formulation
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Inference of Optimality Level
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Synthetic Validation Details
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Global Hyperparameter Table
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S.A EXTENDED RELATED WORK
Here we extend the discussion on some closely related work, showing how our framework differs.
MMynarski et al. 22021) present work related ours- framework construct normative- theory informed parameter priors for parametric models. They specify their normative prior numerically using a specific utility function (O)) for a hand-crafted parameterization in 0. Their prior, exp]SU[(0]
2(3)
crucially requires computing a normalization constant Z by integration, which they estimate by monte carlo but, as they note, is intractable for even 1000 dimensional cases (e.g. our RGC case). Their model comparison application also hinges on this constant (equív. to our marginal likelihood (ML)), and therefore must be approximated. These intractabilities limit the applicability of Miynarski et al's framework small, simple models with well understood, well parameterized normative theories. In contrast, our framework works with any theory that can generate raw predictions (e.g. task-driven networks, autoencoders), as it learns the prior in function space, not parameter space. We are not constrained by dimensionality, and compute our ML analytically. Our framework thus is more automated, general, and scalable while being similarly principled and rigorous.
While we build on the work of Qiu et al. 2023b) and have benefited greatly from their contributions, our work fundamentally differs from theirs in important ways: First, our model is probabilistic, providing well-calibrated uncertainty estimates (Fig. Bc). Furthermore, our normative inductive bias is adaptive to new tasks, and as we show in our analysis of the Occam's Razor effect (Fig. 4p), can even relax its influence as more data becomes available. Additionally, the model of Qiu et al is well designed to support one theory. Their method hard-codes constraints informed by the theory into the first-layer convolutional filters, limiting it to Retinal ganglion cell applications. Our framework's function-space prior is much more general, allowing it to model abstract knowledge that cannot be encoded in a first-layer spatial filter (As discussed in S.C)p. Finally, through its use of exact marginal crmmation-theoreti tool to quantitatively arbitrate
likelihoods, our framework provides principled, between competing and structurally different theories. The closest thing Qiu et al's model can do enable quantitative comparison of learned receptive fields- which, while valuable, is only useful heuristic limited in applicability to this one system.
Other related works The work of Shrinivasan et al. (2023) shares our goal of rigorously testing normative theories with real data. However, our frameworks differ in their core approach and capabilities. Shrinivasan et al focus on manually instantiating variants of a single normative theory (the Neural Sampling Code (NSC)) as probabilistic models. They instantiate the whole generative model hypothesized by NSC P(x, z) and test various competing parameterizations for the constituent P(z[z) and P(x), maximizing the joint likelihoods. Our work is similar in that we also principled instantiate model comparison on the basis of likelihoods of probabilistic models. In contrast their work, we do not hand-pann parameterize and then fit normative- theory inspired probabilistic model, but rather we meta-learn the probabilistic model, represented by the GP kernel function. Thus, the goal of our framework is to enable analysis like that of Shrinivasan et al but for broader class of theories that may be difficult to hand caund-paramtterize as a probabilistic model directly. The scientific value of Shrinivasan's 's work substantiates the need for framework streamlining the process, like ours. Methodologically, Moss et al. (2024) also fit deep kernel GPs on simulated data, but focus on learning solution operator to speed up latent force models, rather than theory-informed kernel for data modeling. Our work is also related to that of Kobalczyk & van der Schaar (2025), who also use Bayesian meta-learning to incorporate human-interpretable domain knowledge into probabilistic models. Unlike us, the source of the knowledge in their informed neural process framework comes from an LLM-generated representation of the knowledge, (e.g. "This plot is increasing"), ", and they meta-learn the relationship between the knowledge embedding and the function space prior itself. Therefore their framework would not be applicable to our setting, as this form of knowledge integration would not help with validating existing theoretical models of neural systems. Finally, while our work is scoped around normative theories in neuroscience, it is related to the technique of "sim-to-real" in robotics (e.g. |Png et al. (2018)) which, similarly to us, endeavors to deploy knowledge learned from simulations to the real world. would be interesting to see if our task-adaptive meta learning approach would help bridge the simulation to reality gap in this domain.
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S.B
EXTENDED FIGURE 3A AND ABLATION STUDY RESULTS
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a
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Informed Deep Kernel
Identity Deep Kernel
Informed Deep Kernel
Random Deep Kernel
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LN Model
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Figure S1: Extended Figure 3a and Ablation study (a) Reproduction Figure 3a. (b) Accuracies of ablation models on biological data.
Identity
RBF
Random
TaskAblation
HcadsAblation
2.84 x10-1
6.56x 10-1
4.34x10-1
8.32x10-1
8.85 1.85x10-1
8
4.39 10
1.89 10 -3s8
2.67x10 24
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4.50 -aa*
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x.0 22
3.37
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32
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1.39x 10 -3
-3e
2.12x 10
2.23x 10
2.77x10
10
64
8.29
6.14x110 33
6.77x10 -3
1.37 3*m
1,05
-7se
7eee
128
1.38x
10 448*
2.35x10 -3*
5.04x10-3a
2.17x10
34*
3.19x10 -55ss
256
6.26x 10 648*
6.84x10 44a*
5.99x10 -3as -as
3.87x10
34*
2.06x10 Gs**
512
4.43x 10 7ame
8.75 10 -iee
4.74x10
2,48x
3
2.98x10
77e*
1024
6.67
10
1.87 10 -5
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44
1.10x10-1
1.15x10- 77ee
1452
7.16x 10 7ame
9,37x -
5.54x10
44
2.47x10-
Table $1: p-values for the one-sided paired Wilcoxon test (Hypothesis: Informed > Control) on the
86 task-level means at each training-set size. < 0.05, < 0.01, < 0.001.
Discussion Figure ST extends Figure 3a from the main text, now including the results from our ablation studies ('Identity', 'Random'; 'Head Ablation', "Tasks Ablation').
The LN model, a common baseline for modeling retinal responses, demonstrates strong performance in the extremely low-data regime. Notably, while the LN model is broadly superior in extremely small data regimes (N < 16), our theory-informed model catches up in performance (16 < N < 64), and subsequently exceeds that of the LN model suggesting that the transferred theoretical knowledge provides an effective bias that helps in data-limited regimes but does not hinder as data becomes more abundant.
Ablation Studies
To isolate the contribution of the theory-derived meta-learned feature extractor (Φ), we performed four ablation studies (Figure STp):
1. An 'Identity' deep kernel ablation: This removes the feature extractor entirely, applying the task-adaptive linear heads directly on input pixels.
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2. A 'Random' deep kernel ablation: This randomizes the weights of & (sampling from the standard initialization distribution), but keeps the architecture (Table fixed.
This ablation represents a powerful comparison target, as large randomly initialized networks, particularly preceding learnable layers like our linear head and RBF GP, have been shown to represent features that capture relevant data structure, preserving imput geometry, and in some cases nearly matching the performance of fully trained features (Kahimi & Recht 2007 Rudi & Rosasco 2017; Vershynin 2018; Saxe et al., 2011 Ulyanov et al. 2018)).
3. A "Task Ablation" condition on the meta-learning process, where we retrain the model, replacing the theory-derived set of meta-training tasks with generic tasks derived from the statistics of natural images alone. We replaced our meta-training set tasks with the tasks of trying to predict the values of the top principal components of natural images- with each task constituting one principal component. We kept all other conditions fixed.
4. A "Heads ablation" condition, where we remove the task adaptive linear heads
Statistical tests We quantitatively compared the performance of our informed model against these two ablations (and the RBF GP) using one-sided paired Wilcoxon signed-rank test (Wilcoxon) 1992) (hypothesis: Informed Control). For each training-set size N,, we first averaged the test correlation over the five random seeds within each task, yielding 86 paired observations for the test (one per task).
The p-values are presented in Table and statistical significance is annotated visually using asterisks in Figure [SIb. Statistical significance (p < 0.05) emerges for all baseline comparisons at N 2 32, with high significance levels (p 10-3) consistently observed for training-set sizes of N > 64. For reference, the median correlation improvement over random features at N = 512 was 0.049 (95% BCa bootstrap confidence interval: [0.018, 0.083), demonstrating that the detected differences are both statistically robust and practically meaningful.
The analysis confirms that our theory-informed model significantly outperforms the standard RBF GP baseline and the other ablation conditions across most training data sizes. The significant improvement over the 'Random' baseline noteworthy; given that randomly initialized deep networks themselves provide strong feature representations (as evidenced by the strong performance of this baseline). The specific features obtained through meta-learning therefore confer a substantial structural advantage. Furthermore, the superior performance of both informed and random features compared to the 'Identity' baseline confirms the significant contribution of the deep feature extractor component itself, beyond the task-adaptive linear heads. Finally the superior performance of the theory-informer meta-learned features over -agoosic meta learned features (designed to encourage good generic features of natural images) shows that the accuracy benefit is specific to the theory-informed tasks.
Take Away: Our ablation study supports the claim made in the main text: the meta-learned features effectively transfer knowledge from the theory, providing beneficial inductive bias that leads to improved performance.
S.B.1 COMPARISON TO TRANSFER LEARNING
We here highlight why we specifically need to use meta learning for our framework, as opposed to simpler mechanisms of knowledge transfer such as transfer learning. In standard Transfer Learning (TL), prior knowledge is encoded as a static feature set. This feature set is then re-used or fine-tuned by simple downstream predictor another task: directly constituting part of the prediction.
Our framework operates on a different principle entirely. While there IS a frozen feature extractor involved, what we transfer is meta-learned theory-informed Bayesian prior over the space of functions. The role of the feature extractor is not to provide part of the solution for a task (c.g. by learning center surround receptive field), but specify intricate function space prior out of an exceptionally vast set of functions representable by an abstract metric embedding.
Intuitively, transfer learning is like memorizing part of the "answer" for one task, and transferring it onto another task. In contrast, our meta learning framework constructs, then transfers a "scheme" for how to find the answer for any task in given domain.
This difference is consequential, conferring the following advantages of our Meta Learning framework over Transfer learning:
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Greater Generality: As prior over functions, method is agnostic exactly -what kind- of knowledge it needs to encode. While for fitting an RGC response to images, one could just well invent clever regularizations accuracy gains, this approach is limited to forms of knowledge that can be expressed as specific neural network weight values/statistics. In contrast, our prior capture more abstract forms of knowledge, for example abstract rules for a rule-based decision making task. We discuss this more Sec. S.C
CNN filters achieve similar or superior
Task-Adaptive Prior Unlike transfer learning, where the knowledge transfer is fixed, our theory informed prior can adaptively relax itself or absorb more informative structure via the linear heads. This enables generalization from idealized efficient coding predictions to the real, noisy biological data.
Principled Theory Validation and Uncertainty Quantification In our informed GP, we can compute the exact marginal likelihood of biological data under the theory-informed prior- direct measure of the evidence for given theory in units of information. This enables a direct quantitative comparison between competing scientific theories. Our GP is also probabílistic: presented with image that its training data is not representative of, it outputs a high degree of uncertainty (Fig 3b,c), and not just a confidently wrong answer.
Transfer Learning Baselines We ran extensive grid search to fine-tune linear and MLP heads on top frozen features from pre-trained ResNet-18 models (using both ImageNet and SimCLR trained weights). As shown in our figure, performing worse than even the plain MLP on the raw data (Figure S2)
found that these pretrained features were not competitive,
We hypothesize this is because the features from these deep networks are too abstract for RGC responses, a finding consistent with prior work ( (Cadena et al.] (019)) which found earlier-layer features were necessary even for modeling V1.
Intietrdd Kermst Uinar ShuC1
0.25
Test Accuracy
8:15
0.05
128
256
512
1024
142
Train Set Siza N
Figure $2: Transfer Learning baselines
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S.B.2 ANALYSIS OF RELATIVE SOURCES OF INACCURACY OF DIFFERENT MODELS
b
Spearman between Accuracies
0.5
TIK
1.0
PCA Ablation
RBF
Spearman p
0.8
Test Accuracy
0.6
0.4
0.24
0.2
0.0
TIK and RBF
TIK and PCAAblation
0.1
10 Lowest
10 Highest B*
B*
Figure S3: Analysis of sources of error a) Depicts Spearman rank correlation of accuracies of different methods across the 86 biological neurons, averaged over seeds. b) Depicts accuracies of TIK, PCA ablation, and RBF GP's on the 10 biological neurons with the lowest and highest 3 scores by BMC procedure (Fig.) 5)
Here we analyze what differentiates biological neurons on which TIK performs well from those where it performs poorly. We first compare per-neuron prediction accuracies (correlation the test set, averaged over seeds) between models. As shown in Fig. S3h, the accuracies of TIK, the PCA ablation, and the RBF GP are strongly rank-correlated across neurons (e.g. Spearman p R 0.77 for TIK vs. RBF). This indicates that all three models tend to succeed and fail on the same neurons, suggesting that substantial component of the performance variation across neurons shared and not specific to any one kernel. We next stratify neurons by their inferred 3 from the TIK-RBF mixture (Fig. 5p). Fig. S3p shows the mean test accuracy of each model on the 10 neurons with the lowest and highest 3 On the 10 lowest-8" neurons, all three models perform similarly poorly (mean Pearson r z 0.13 for TIK. PCA. and RBF), indicating that for these cells neither the EC-informed features nor the alternative deep RBF kernels provide a strong inductive bias. In other words, 33 R 0.6 corresponds to neurons that are difficult to predict for all models, rather than cases where the RBF null alone is good fit. In contrast, all models but in a graded fashion: relative to the low-3" group, the RBF GP improves by ~ 0.16 Pearson the PCA ablation by 0.24, and TIK by 0.32. Thus, neurons that our BMC procedure features
the 10 highest-9" neurons, accuracies increase for
classifies as better explained by the theory are precisely those for which Ancory-informa yield the largest additional predictive gain. The fact that the PCA ablation sits between RBF and TIK is consistent with the picture that generic natural-image structure (captured by PCA) explains part of the improvement, but that the full gain for high- neurons only realized when using features meta-trained on the specific efficient-coding normative model.
Implications. Shared difficulty across models. The high rank-correlations in Fig. S3a show that neurons which are hard for TIK are also hard for the PCA deep kernel and the RBF GP. This suggests that the dominant source of performance variation across neurons intrinsic unpredictability of the recorded responses (due to biological variability and/or measurement noise), Role of efficient-coding structure. Conditioned on neurons where structure is more easily discernible (high ff*), we observe graded improvement RBF < PCA < TIK. This pattern indicates that, beyond generic natural-image statistics, the EC-derived feature space provides additional, neuron-specific structure that improves predictive performance. Interpretation of f High 3* can be interpreted identifying neurons whose responses are especially well captured by this efficient-coding solution, whereas 3 z 0.5 flags neurons for which (i) none of the tested kernels (TIK RBF) provide strong advantage over one another for explaining the data (c.g. due to noise), or (ii) the structure truly lies in between the two (e.g. in synthetic, noise free example)
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S.C ON THE GENERALITY OF OUR FRAMEWORK
While our manuscript presents single challenging deep-dive case study of the retina application, our framework is, by design, quite general.
S.C.1 DEEP KERNELS DEFINE BROADLY APPLICABLE FUNCTION SPACE PRIORS
The generality of the Deep Kernel architecture stems from core principle: many complex biological phenomena are governed by a small number of simple, underlying latent variables. Systems like these can be well described by our model.
To elaborate, metric-based methods confer significant flexibility when used to define function space priors in deep kernel Gaussian process setting. Since the core of our model is a Gaussian Process with learned kernel (x, x') = of expl- (x (e(2")))1 DII""
the prior over functions is fundamentally defined by the geometry of the embedding space learned by z and refined by h,
21
The core assumption of this architecture is that the deep neural network can learn a transformation of the potentially complex input space X' to embedding space Z where the functional relationships become simple, smooth, and stationary. The RBF kernel enforces strong prior for smoothness in close in Euclidean distance, their
this learned space Z: if two points (x))) and he(()(x')) corresponding function values ((x and f'(x') are modeled to be highly correlated.
Therefore, the space of priors that can be fruitfully expressed is the set of all functions that can be simplified to a stationary smoothness assumption through a continuous mapping. This architecture is naturally suited for problems where the data generating function fág(2) : R E exhibits specific structure. The hypothesis is that while fäg may be complex in the high-dimensional input space X. its structure is fundamentally governed by a smaller number of specifie continuous latent variables, More formally, we assume there exists oowddimensional manifold M parameterized by latent coordinates is that the true function can be expressed as the composition of "simple" function fsimple Z + R and the inverse of the parameterization, g fag (ar) = simple (g- (I)) for MM)
(where dim())<dim(²)) via a map g V + M C X. The core hypothesis
The role of the deep network and linear head is to help approximate this g_ allowing our Ssimple (an RBF Gaussian Process) to be good prior for the data generating function.
This is closely related the manifold hypothesis in machine learning (Bengio et al. 2013 Calandra al. 2016), and is extraordinarily general across natural science datasets. We provide numerous examples in Table S2
S.C.2 GENERALITY VIA A MODULAR META-LEARNING PIPELINE
The generality of our work stems not only from the deep kernel architecture but from the underlying meta-learning pipeline itself. The framework's components are modular and can be adapted to problems of varying complexity and data modality.
Adapting the Feature Extractor Data Modality: While we employ CNN for image data, the feature extractor vi is pluggable module. For problems with different data structures, it could be replaced with more suitable architecture, such as a Graph Neural Network for molecular data or Transformer for sequence data, without changing the overall framework.
Adapting the Kernel to Problem Complexity: The choice of a deep kernel motivated by our complex. high-dimensional problem. For simpler scientific problems defined on low- low-dimensional inputs (c.g., modeling ID orientation tuning curves), a deep feature extractor would be unnecessary. In such cases, framework could be instantiated by meta-learning the hyperparamcters of an expressive classical kernel, such as the Spectral Mixture Kernel (Wilson & Adams) (003)), directly from theory- tasks. This demonstrates that the core principle- principle-meta-leammee meta-learning prior from theory-is broadly applicable across different model complexities.
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System Model
Input
Output y
Latent Vari-
(Normative)
able V
Theory
Primary Visual Cortex (V1)
Images
Single Neu- ron rates
Stimulus
Sparse
firing
features
Coding
(Ol-
(orienta-
shausen
tion, spatial
Field 1996)
frequency.
motion) Intended Limb Kine- maties (Position, Velocity, Ac-
Primary Motor Cortex (MI)
Neural Popu-
Limb Veloc- ity vector
Optimal
lation Activ-
Feedback
ity
Control
Todorov &
Jordan 2002
celeration)
Vargas al.
2024)
Primary Auditory Cortex (A1)
sound's
Single Neu- ron firing rate
Spectro- temporal features like pitch, timbre
Efficient Coding Natural Sounds
time- frequency spectrogram
of
Lewicki
2002)
Hippocampal Place Cells
Multi- Sensory inputs
Place Cell fir- ing
The animal's 2D spatial position, proximity to abstract goals (El-Gaay et al.
Successor
Represen-
tations
Momenne-
jad et al.
2017)
2024
and
head
direction.
Prefrontal Cortex (PFC)
Stimuli and Reward his- tion activity tory
PFC popula-
Abstract Task Variables ("belief state", "deci- sion rule")
Bayesian
Decision the-
ory (Körding
& Wolpert
2006). Drift
Diffusion models Molecular
Protein Science
Amino Acid Sequence
A functional property (e.g., catalytic effi- ciency, stabil- ity) reactivity (EC50)
Geometric or electrostatic properties of the protein's active site pharmacophores DFT QSAR pre- dictions
Dynamics
simulations
Molecular Property prediction
Small
or
Molecule
or structural motifs
features
(Connectiv-
ity Finger- print) Vector of local mor- gene expres- sion level
Developmental Biology
Cell-fate
Cell's physi- cal position along de- velopmental
Reaction-
Diffusion
phogen concentra-
Models
Zagorski
tions
axis
et al. 2017)
Table $2: selection of scientific systems where the core hypothesis of our framework may be applicable.
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S.D BIOLOGICAL DATA PREPROCESSING
S.D.1 DATASET SOURCE AND INITIAL PROCESSING BY QIU ET AT..
Our dataset was sourced from the published work of Qiu et al (Quu al. 2023b a). focusing specifically on the responses of 86 mouse ventral ganglion cells to a 30Hz natural movie stimulus with Green and Ultraviolet (UV) channels, described in main text. To reiterate, each input regression model is derived from the downsampled image frames (36x32 pixels) presented at 30Hz to the retinal cells. The outputs we predicted are the Calcium fluorescence responses measured for each neuron, acquired via two-photon microscopy at frame rate of 7.8125 Hz. These fluorescence signals serve as proxies for neuronal firing activity, as calcium indicators reflect intracellular calcium concentration changes associated with cells firing action potentials. While we applied our own our processing steps (temporal flattening via a canonical filter derived from an LN model, hierarchical clustering to select unique representative images, selection of the UV channel, and 7-scoring), Qiu et al. performed crucial initial preprocessing on the data, most important of which we describe here- but refer readers to their publication for full details.
Cell Selection via Quality Index (QI) The dataset includes responses from cells selected based on their response reliability. Qiu et al. calculated Quality Index (QI), ranging from 0 to 1, for each cell based on the consisteney of its responses across repeated presentations of specific test sequences. The QI was calculated as:
Var (E (C)
QI
E, (Var, (CT)
where C' represents the response matrix with dimensions time samples (t) by repetitions (r), Ey1- denotes the expectation (average) across repetitions, and Var, denotes the variance across time samples. higher QI indicates a more reliable response with a higher signal-to-noise ratio. For inclusion in the final dataset used for modeling, neurons responding the natural movie stimulus had to meet the criterion Q > 0.25, where Tonse was determined using responses to separate noise stimulus test sequences also presented during the experiments.
Stimulus and Response Handling Importantly for evaluation, the 750 test and validation fluo- rescence responses provided in the dataset are not signals but represent average across repetitions of specific held-out input sequences from the natural movie stimulus. This averaging serves to increase the signal-to-noise ratio of these benchmark traces. This makes the original train/test+val split non-interchangeable.
For exhaustive details the experimental procedures (including animal handling, tissue preparation, recording setup) and the complete preprocessing pipelines applied across all stimulus conditions in the original study, we refer the reader the source publications (Qiu et al. 2023b a). Our subsequent processing steps applied for this work are detailed below.
S.D.2 TEMPORAL FILTER EXTRACTION VIA LN MODEL
To convert the regression problem from movie inputs image inputs, we temporally flatten the movies. To achieve this, we first fit a spatiotemporal Linear-Nonlinear (LN) model to predict the 16200 neural responses E IR 16200x88 from 16200 movie inputs e I 162000 consists of spatial filter parameters W E S6111152 and Temporal filter parameters E 86 Among temporal filters we observed classic biphasic shapes (dip-then-peal or vice versa). We then "flipped" some filters by multiplying by -1 in order to align their biphasic structures and allow for averaging, and then averaged across neurons to obtain a single "canonical" filter F E R030, spanning 50 frames. The choice of which filters needed flipping was made by thresholding the peak value. Each movie frame 2, E x32 was then projected onto this canonical temporal filter, effectively collapsing the time dimension into a single static "image", . Σ50 F,K,, We plot all 86 time filters identified this way (Fig. S4 alongside the canonical average filter.
This LN model
S.D.3 FILTERING NON REDUNDANT IMAGES
To derive a representative subset from the 16,200 images in our dataset, we applied a data-driven hierarchical clustering approach. We first computed 1-NN distances (excluding self-matches) in the
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Temporal LN Filters
3
Individual Filters
Mean Filter
Filter Amplitude
20
50
Time Step
Figure $4: Time Filters of LN model on Biological Data Individual filters shown are post-flipping
feature space (shaped 16200 x 1152) using Scikit-Leam's NearestNeion ghbors ghhborrss with Euclidean distance. These distances sorted, and clustering threshold was selected based the largest gap between successive values a heuristic for natural cluster separation. We then performed agglom- erative clustering with average linkage and this threshold (with u_cllusters==one) resulting in 1,452 clusters. A single exemplar (the first image) from each cluster formed our representative subset.
S.E SYNTHETIC META- DATASET GENERATION
This section outlines the generation of the synthetic dataset based on our chosen efficient coding theory.
S.E.1
NORMATIVE MODEL (EFFICIENT CODING AUTOENCODER)
Architecture of Efficient Coding Autoencoder
Table S3 shows the full architecture used.
Layer
Details
Output Shape
Activation
Input
B, 1, 28, 28
Conv
9xx, padding=4
B, 16, 28, 28
ReLUU
Flatten
B, 12544
Linearl
12544 - 300
B, 300
Gaussian Noise
2.0 (train only)
B, 300
Bottleneck
B, 300
ReLU
Linear2
300 - 12544
B, 12544
ReLU
Unflatten
B, 16, 28, 28
Conv ConvTranspose
9x9, padding=4
B, 1, 28, 28
Tanh
Output
B, 1, 28, 28
Table S3: Normative Autoencoder Architecture
Training Procedure The autoencoder model, adapted from work by Ocko et al. and Qiu et al. (Ocko et al. 2018: Qiu et al. 20236), was trained to learn an efficient code for natural images. The core idea of efficient coding is that sensory systems aim to represent naturalistic stimuli faithfully while minimizing metabolic costs. Our autoencoder operationalizes this by optimizing the loss function LEC: defined as:
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LEC = MSE X X) + Abattoaateccc//1 + a (I] /owe 112 + Teovv
(S1)
where MSE(X, N ソ ||x, -X51lž is the mean squared error ensuring fidelity between original (X,) and reconstructed (X,) images over batch of N = 200 samples. The second term applics an LI penalty to the bottleneck layer activations Abortenckk with weight = 1 x 10 to promote sparse codes. The third term imposes L. penalty the weights of convolutional cow) and transposed convolutional wwm)) filters with weight a = 5 x 10-5 to encourage smooth filters. Optimization performed over 200 epochs using the Adam optimizer (learning rate x 10-5). We trained on 60000 patches sampled from the public Van-Hatered dataset of natural images (Van Hateren van der Schaaf 1998).
Consistent with efficient coding frameworks that account for noise, additive Gaussian noise (std 2) was applied the input images during training, simulating sensory noise and acting a regularizer. While Ocko et al. (Ocko et al. 2018) primarily used noise and sparsity penalties without an explicit narrow bottleneck in their model, our implementation imposed fixed bottleneck size. To ensure we were meta-traine ceta-training on a high-fidelity representation of the theory, we performed a small hyperparameter search over the normative model's architecture and selected a 300-unit bottleneck, along with the other hyperparameter values. This was pragmatic choice that reliably produced the canonical center- surround receptive fields predicted by efficient coding. Our chosen hyperparameters (a, B. input noise, bottleneck size), represent one point in a large trade-off space between reconstruction accuracy. code sparsity, and filter smoothness, and were chosen to ensure a robust and faithful instantiation of the theory for our subsequent meta-learning. Varying these parameters would likely yield different efficient coding solutions, resulting in different filter properties (e.g. center/surround relative and absolute size) as the model adapts to different balances of these competing objectives under varying architectural and noise constraints.
Example Receptive Fields To validate the features learned by the hidden units in the autoencoder's bottleneck layer, we visualized the preferred input pattern associated with each unit. This was achieved by activating each of the 300 bottleneck units individually (setting its value were 0) and passing this sparse representation through the trained decoder network. The resulting decoded output image represents the spatial pattern that maximally drives (or is reconstructed by) that specific bottleneck unit (Fig. (5)
while others
Figure S5: Normative Autoencoder Receptive Fields
S.E.2 DATA AUGMENTATIONS PERFORMED TO EXTEND THE META- TITA-TRAIN SET
To generate the synthetic "tasks", we extract spatial receptive fields the "optimal" bottleneck neurons of our efficient coding autoencoder by fitting a spatial Linear-Nonlined (LN) model the
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activations of the autoencoder bottleneck. This yields 300 center-surround filters ft Each filter then defines one synthetic task obtained by taking its dot product with the image stimuli yi = f I where F& is the k'th image. We expand this meta- set by choosing 20 archetypal un-obfuscated center surround filters (by visual inspection), rescaling their size (randomly ranging from 0.8x to 1.2x) and jittering the location of their 'center'' allowing us to increase the size of the synthetic meta-train set Nr. We created 4000 such synthetic tasks, but ultimately used only N = 490 selected randomly, as we found empirically that this number of tasks saturated the meta- titta-training performance.
Rationale behind Jittering and Scaling, Our Jittering and Scaling augmentations are not arbitrary, but rather principled computationally efficient way to generate new tasks consistent with the same normative theory. Our meta learning framework treats each neuron as an individual "task". meaning our framework learns a prior for single-neuron responses. Our meta-trai veta-train set is therefore composed of many separate individual examples of "good" single neuron solutions consistent with the theory which is itself posed a population level. On single neuron level, efficient coding theories predict type of solution (a center-surround structure), not one single filter at one specific pixel location. Therefore, a filter jittered to a different location is an equally valid example of nigh--performing solution under the theory of Ocko et al: it would have effectively the same utility level. This is in stark contrast to population level efficient coding theories, which may precisely specify the relative positions, overlap levels, and tilings of population of efficient coding neurons- so if we were learning a population-level prior rather than a single neuron level prior, the jittering would be problematic as it would be deleting that information.
The spatial scale of receptive fields, on the other hand, is a known consequence of architectural capacity and regularization in efficient coding models. Ocko et al, for instance, have analytically connected the size of the emergent receptive fields to the "stride length" of the convolutional filters- driven by the need to prevent aliasing. Other seminal works, such as that of Karklin and Simoncelli (Karklim& & Simoncelli 2011), have demonstrated that the "optimal" efficient receptive field size is a function of the input noise level. Because the noise is applied i.i.d to each pixel, a high level of ratio of high-frequency components of the image's 's power spectrum. Therefore, in a high-noise environment, the optimal solution is to average over a larger spatial area to filter out the noise. This corresponds to larger receptive field. On the other hand, given low noise, the high frequency features are more reliable, and therefore valuable to capture with spatially smaller receptive fields.
noise primarily affects the Signal-to-noie
There is not a single efficient coding theory predicting a single "optimally sized" canonical receptive field. The optimal size is conditional the stimulus statistics and architectural constraints. In fact, biological retinal ganglion cells are known to adapt the sizes of their functional receptive fields via inhibitory plasticity precisely in response to these changing environmental statistics (Smirnakis et al., 1997). Therefore, our choice to spatially scale the sizes of receptive fields in a reasonable range is a prinicipled, computationally efffciint way to convert our Efficient Coding solutions trained at single fixed noise level and architecture into a family of solutions consistent with a larger range of these constraints- which are in turn all consistent with the theory- being optimal solutions corresponding to different hyperparameter values.
If we had known the precise noise levels experienced by the photoreceptor cells of the ex vivo biological neurons, we may try to reverse engineer a single noise level in our model consistent with their true underlying noise level and test the predictions of the theory at that level of specificity. However, since that is far beyond our scope, scaling the receptive fields the most principled thing to do: resulting in a set of meta-train tasks representative of a wide range of these hyperparameter values which we as scientists do not know the ground truth value of.
Nonetheless, we provide an Ablation study on the Scaling data augmentation, demonstrating that model's accuracy is, if anything. reduced by the preprocessing across most data sizes.
N
4
16
32
64
Theory Informed Kernel Ablation: No Scaling
0.015 = 0.015
0.072 + 0.019
0.135 + 0.017
0.182 = 0.019
0.217 + 0.014
0.021 + 0.015
0.079 + 0.022
0.154 + 0.020
0.192 + 0.022
0.231 + 0.010
N
128
256
512
1024
1422
Theory Informed Kernel
0.250 + 0.005
0.287 + 0.004
0.307 + 0.005
0.323 + 0.004
0.325 + 0.002
Ablation: No Scaling
0.256 - 0.008
0.289 + 0.007
0.305 + 0.005
0.314 + 0.002
0.324 + 0.001
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S.F META-LEARNING ALGORITHM AND TRAINING DETAILS
S.F.I MODEL ARCHITECTURE
As general principle, tried to minimize the flexibility of our task task-adaptive components, as doing so in conjunction with bilevel optimization may reduce pressure on the meta learned feature extractor to learn during meta training- since the task adaptive components would flexible enough to capture the requisite structure on their own.
Feature Extractor Architecture
Table S4 shows our feature extractor architecture.
Layer
Details
Output Shape
Activation
Input
B, 36, 32
Conv1
3x3,
padding=1
B, 32, 36, 32
GeLU
Conv2
3x3. padding=1
B, 64, 36, 32
GeLU
MaxPool
2x2
B, 64, 18. 16
Conv3
3x3, padding=1
B, 128, 18, 16
GeLU GeLU
Conv4
3x3, padding=1
B, 128, 18, 16
Flatten
B, 36864
Linearl
36864 + 256
B, 256
GeL.U
Linear2
256 -> >256
B, 256
Table S4: CNN feature extractor (δ) architecture
Linear Heads All linear heads are torch. nn Linear instances, with 256 input dims and 128 output dims and L1 regularization (=0.01) to encourage generalization (Fumero et al. [2023), also following the recommendation from (Van Amersfoort al. (02T) not overly shrink the latent dimensionalities.
Gaussian Process Layer For all of our GP layers, we used GPyTorch Exact GP with an RBF
Kernel =' o exp(- exp(-Ila=s' - We all mean functions ZeroMean () and used Gaussian Lengthscale Prior with variance of 0.01 centered around the initialization lengthscale value. For initialization, we followed the heuristic "median lengthscale init" reported previous work (e.g. (Van Amersfoort et al. 2021) where the lengthscale is initialized to be the median distance between the embedded data points. Importantly, we did not keep re-computing the median init as we cycled through task batches, instead saving the first computed median and re-initializing with that value. Experimental validation demonstrated the importance of this fixed median initialization and narrow lengthscale hyper-prior for avoiding feature collapse (Sec. S.F.4
S.F.2 META-TRAINING PROCEDURE
Table S5: Hyperparameters for the Optimization Procedure (Algorithm)
Hyperparameter
Symbol
Description
Value
Number of Epochs
E
Total number of meta-tree meta-ttaining epochs.
7
Batch Size
B
Number of tasks per meta-ban batch.
10
Inner Adaptation Steps
Tinner steps
Number of gradient steps for task adaptation.
17
Inner Learning Rate (Linear)
Olinear
Learning rate for linear head parameters in inner loop.
0.0004
Inner Learning Rate (GP)
agG
Learning rate for GP hyperparameters in inner loop.
0.002
Outer Meta-Update Steps
Touter steps
Number of outer loop optimization gradient steps Learning rate for meta meta-parsmeters (fmeta) in outer loop.
3
Outer Learning Rate
Oquuter
0.0004
Our training procedure (Alg. I is similar to that of the "Adaptive Deep Kernel Fitting via the Implicit
Function Theorem" (ADKF-IFT) method by Chen al (Chen et al. 2023). We employ Bi-level
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We trained on batches of 10 (synthetic) tasks time, each task containing 1452 labeled input/output training data, randomly sampling 49 such batches, resulting set. We trained for 7 epochs in total. At the start of every epoch, we ran the full adaptation procedure on three meta sets, using 100 support points each. One on Synthetic data from the meta train set. One on Synthetic data unseen during meta training but from the same distribution. One our Biological data, with all evaluations computed on the validation data only, using a subset of tasks. We employed early stopping based held-out biological data Pearson correlation (Fig. (S During our training process, we clipped the gradients on the base feature extractor to have a maximum norm of 1, and we trained the first epoch with 1/10th the learning rate. As shown in Alg. D we optimized the feature extractor parameters Own on the log-probability of the query data (outer level) in the meta-train set and Published as a conference paper at ICLR 2026
Algorithm Procedure for Fitting Adaptive Deep Kernel Model
1: Define: Task-adaaptive parameters Outge (i.. Oep, W. from h.)
2: Define: Metn-learned parameters Paxu (i.c. Pax from
3:
Input: Mcta training act
T [Ti (Diam ) )
where for each task
and D ((a7,)) que2 quan
D.
[111,
4:
Shuffle T and partition into batches for opoch = to do
5:
6:
for butch E. 'I do for cach task T; = (D. D in the batch do Reinitialize task-adaptive parameters Osapes
7:
8:
9:
for r to Piass do Update task-spocific parameters:
10:
Bstope. Oscape, Olnnne datase log P (unoot P(ymuoo 22:00
Beogee a Oriza
where Ctinner end for
(Airrears agg)) denotes the learning rales for the lineur head and GP hyperparametters, respectively.
11:
12:
Set 9rigee Ba
13:
end for
14:
for to Plvsser cxp. do
15:
Update the meta parameters:
Ouens t- Ousts
Ceotter V Prass Pichash log P ((v 7 m pirrom Cmm, Ouru)
where Ouone is the meta-lcariin rate.
16:
end for
17:
end
18:
end for
optimization: with disjoint adsk-tde and meta-learned modules trained separately, different learning objectives. Like Chen et al, we use the standard GP marginal log likelihood (Alg. Eqq. 1, main paper) for our "inner" training loop task-adaptive parameters and the log-probability of the query data for the "outer" training loop on meta-learned parameters (to encourage generalization enrich the outer level gradient, as this would not be computationally feasible with our significantly higher number of adsk-adaetive parameters. Future work may explore the likely benefits of employing a scheme to approximate these implicit gradient terms (Geng et al.] 202T). We also make adjustments to the training process to avoid feature collapse, discussed in S..F44
(Loth et al., 2022 ). Unlike Chen et al, do not use implicit differentiation
total of 490 tasks in our meta train
the linear heads/ GP layer parameters (W; and gg..) on the MLL of the support set (inner level). For synthetic meta train sets, we manually set the GP noise level close to zero to avoid feature collapse- preventing the log-prob objective from hiding collapsing epistemic uncertainty behind high alleatoric uncertainty (as discussed in (.F.4... Our experiments indicated that utilizing smaller proportion of data points for the support set, with a correspondingly larger proportion for the query set, fostered more effective representation learning in the feature extractor. In our case, out of 1452 possible points to be divided into both for each task, we used 5% of them for our support set, and 95% for Query set, forcing the model predict more unseen points using fewer conditioned values, and re- sampling the exact support/query split every epoch. We also adhered to the torch. double dtype avoid numerical errors from the low noise level.
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1.0
3
RMSE (support)
NLPD (query)
RMSE (query)
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NLPD alleatoric
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Pearson Corr
NLPD
meta-RMSE-In
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meta-NUPD-in
RMSE
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metaa-cor-bboooogcalaaalll
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Figure S6: Training Ltats,Left Plot, Left Axis corresponds to mean RMSE (averaged over tasks), right axis corresponds to pearson correlation averaged over tasks (biological). Right plot depicts NLPD values, with NLPD(query) depicting epistemic NLPD and NLPD alleatoric including the noise term (which here set to be very low). Dotted lines signify post-adaptation tests of the learned feature extractor at the start of each epoch. Solid lines are stats accumulated during training. The dotted lines are evaluated at the start of every epoch, while the solid lines are evaluated at the end of every epoch, which is why there is one extra dotted line point. Meta RMSE and Meta NLPD correspond to unseen tasks drawn from the same distribution, and meta-rmse and meta nlpd in correspond to tasks from the meta train set. To be clear, early-stopped using the dotted purple line, which reflects what a scientist using our method should/would do.
S.F.3 TASK ADAPTATION PROCEDURE
For each meta-test task (i.c., each biological neuron), we perform task adaptation to fit the model to the available biological data (the support set for that task). This involves optimizing only the task- specific parameters while keeping the meta-learned feature extractor weights frozen. The parameters adapted for each task i include the weights of its linear head (W₂) and the hyperparameters of its Gaussian Process (GP) layer 0gp ap... f 4 li, on.)), encompassing the GP output scale, length scale, and likelihood noise variance, respectively.
Adaptation is performed by maximizing the marginal likelihood of the support data for that task using the Adam optimizer (Kingma Bal 2014) for 300 epochs. We used a base learning rate of 4 x 10-3 for the GP hyperparameters and Adam betas of (0.99, 0.999). The learning rate for the linear head weights W. was scaled down by a factor of 10-2 relative to the base rate (i.e., 4 x 10-9). Additionally, an I.1 sparsity penalty with coefficient of 10-2 was applied to the linear head weights during adaptation. For the GP hyperparameters, the length-scale prior was kept narrow (Gaussian with variance 0.01, except for the RBF-only case) and its mean was initialized using the median heuristic applied to the pairwise distances of the features d(z, x') = || (é())) h, (ø(x')) from the frozen extractor and freshly initialized heads. For experiments involving synthetic meta-test data, the GP likelihood noise Jyss was initialized at 10-4, reflecting known ground truth. For biological data, the standard GPyTorch initialization was used.
Crucially, these same adaptation settings (optimizer, learning rates, epochs, priors, initializations) were used consistently across our main "Informed' model and all ablation conditions. Given the significant computational cost required for extensive hyperparameter tuning for each model variant across dozens of tasks and multiple data subsets, we adopted these fixed, reasonable settings based on preliminary experiments. The consistent application ensures fair comparison between the conditions, and the sensible trends observed in the results (e.g., Fig. [SI) suggest that these settings, while perhaps not perfectly optimal for every single task/model pairing, were adequate and did not unduly handicap any specific model.
S.F.4 DESIGN CHOICES MITIGATING FEATURE COLLAPSE IN OUR DEEP KERNEL MODEL
Although literature such as (Chen et al. 2023 suggests that meta-learning across diverse tasks might inherently mitigate feature collapse, experiments indicated this was guaranteed for our
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case. However, we successfully avoided feature collapse in model, attributing this primarily to three specific design choices in our training methodology:
1. Narrow Hyperprior on GP Lengthscales: We imposed tight prior distribution on the Gaussian Process (GP) lengthscale parameters (4z), severely restricting their ability adapt during the inner-loop optimization for each task.
2. Fixed GP Lengthscale Initialization: We did not re-initialize the GP lengthscales using the median heuristic at the start of each task batch adaptation, a technique commonly used to set reasonable initial values (e.g.. Chen et al., 2023) Van Amersfoort et al. 2021). We used the median heuristic to compute the lengthscale once, the start of the first task batch, and then used that lengthscale throughout the other taskbatches.
3. Epistemic Log Probability Objective: Our outer-loop meta- veta-training objective optimized the feature extractor (Φ) based on the log probability of the query data (3/y) conditioned on the support data ((r,,,+)) ]). often referred to as the negative log predictive density (NLPD). Unlike existing training approaches, however, we explicitly used the log probability of the data excluding the likelihood noise. This focuses purely on the epistemic uncertainty captured by the GP, unlike objectives combining epistemic and aleatoric uncertainty (e.g..
Chen et al. 2023 Lotfi et al. 2022) optimizing the log marginal likelihood (LML) directly (e.g.. Patacchiola et al. 2020).
Our preliminary understanding of the mechanisms by which these choices prevent collapse follows:
Choices 1 and (Lengthscale Restriction): We observed empirically that allowing the GP length- scale (1,) to be freely optimized within each task's inner loop (as is typical in GP regression and done for task-adaptation
(Chen et al., (023)) or frequently re-initializing via the median heuristic, created detrimental positive feedback loop during meta-training. Shorter lengthscales appeared to incentivize the deep feature extractor (Ø) to map inputs closer together in feature space. This proximity, in turn, further encouraged shorter lengthscales via the inner-loop LML optimization, ultimately leading to collapsed feature representations where distinct inputs map to nearly identical points. By fixing the lengthscale via narrow prior and consistent initialization, we broke this feedback loop. The potential loss of flcxibility from a fixed lengthscale was compensated by the adaptability of the meta-learned feature extractor (e) and the task-specific linear heads (h,), which could still arrange the features appropriately for the given lengthscale.
Choice 3 (Epistemic Log Probability Objective): This choice appears to counteract feature collapse through two mechanisms:
Avoiding Prior Log-Determinant Minimization: Ober et al. (Ober et al. 2021) identified key mechanism driving feature collapse in DKL, models trained via LML maximization. The LML. objective can be decomposed into data fit term and complexity penalty term:
1 3 ((K + + NN y log K + o INI
N 2
log
log(2r))
2
Data Fit Term
Compleity Penalty Term
Crucially, Ober et al. prove (in their Proposition 1) that for kernels with learnable output scale (a y), the data fit term becomes a constant (-N/2) at the optimum. Consequently, maximizing the LML with highly flexible feature extractors becomes dominated by minimizing the complexity penalty, specifically the log determinant term log K + N. The model achieves this by learning feature mappings (Ø) that make the kernel matrix K (evaluated on the learned features) as close to singular as possible, which corresponds to collapsing the features.
Our objective. however, focuses on the log probability of query data My given support data Us), which depends on the posterior predictive distribution for the latent function values fa at query inputs Xg. This distribution is Gaussian, p(ff, Xx,, X,, Vs) = (ff Egja). Let x.= (- ... F.,n. be the set of n. support inputs and X, { Fgg } be the set of ny query inputs. The kernel function, parameterized by the feature extractor o, is ka(:)). We define the following kernel matrices:
K,s [ko (e the ny ny kernel matrix between support points.
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Kg 1ho qq.s Zz 12. the ny x mg kernel matrix between query points.
Kgs (T L Mi=1j=1. the ny x n, kernel matrix between query and support points.
Kg KTT qas the ny x ng kernel matrix between support and query points.
The posterior predictive covariance is then given by gg1= = Kg (g + IN, Kgg where oR is the observational noise variance and T,., the n, ns identity matrix. The relevant term in our objective related to model complexity is - log -- By properties of determinants of positive definite block matrices (specifically, the joint prior covariance matrix over fa and fq). it follows that | S K Therefore, log | I log / and minimizing log [Zigla] as part of our objective is a different optimization target than directly minimizing log gg We hypothesize that this difference mitigates some of the detrimental effect of minimizing the log determinant of the prior covariance (i.e., log gg or terms in log mode studied by Ober et al. Notably, the outer meta-learinng loop objective used by (Chen et al.) 2023) is also based on the query set predictive posterior, suggesting they might also have benefited from this effect.
K2. o I,, D. reducing our susceptibility to the failure
Explicit Epistemic Uncertainty Optimization: A more speculative reason relates to the nature of the objective. Feature collapse in deep kernel GPs affects the uncertainty of their latent function values. Therefore, optimizing for an objective that explicitly depends on the latent function uncertainty (i.e., epistemic uncertainty via Eggee) may have encouraged the formation of features that avoid collapse. Specifically, overconfident latent function predictions associated with feature collapse (i.e., near-singular Eg(a)) would be heavily penalized by the data-fit component of the loss function (which involves E. unless predictions were unrealistically perfect. This leads to a poor overall log probability. If we had trained
objective incorporating observed (aleatoric) noise, sufficiently large learned noise variance (elatoore)) could stabilize the predictive covariance (i.e., in terms like Egls + o Meeaoone thereby masking the collapsed epistemic uncertainty Egis and reducing the direct pressure on the feature extractor to maintain well-calibrated epistemic uncertainty.
S.G IMPLEMENTATION DETAILS AND RUNTIMES
S.G.I SOFTWARE AND HARDWARE
All models were implemented and experiments were conducted using Python 3. 3.11.12. The primary deep learning library utilized was PyTorch 2. 6. 0+cu124 (Paszke) 2019). Gaussian Process functionalities were implemented using GPyTorch 1.14 (Gardner et al. 2018). Other key libraries included NumPy 2.0 2, SciPy 1. 1.14... and Scikit-learn 1. The experiments were predominantly run on Google Colaboratory (Colab). The primary compute hardware utilized for experiments
single NVIDIA 14 GPU.equipped with approximately 23 GiB of VRAM (23034 MiB). We acknowledge the use of Large Language Models (LLMs) to assist with generating boilerplate code, suggesting edits, and aiding debugging tasks during the development process- all strictly reviewed for correctness.
S.G.2 BASELINE AND ABLATION MODEL DETAILS
We implemented CNN baseline by exactly replicating the "Systems Identification" branch architecture presented in (Qiu et al. 2023b), selected because it represents well-validated model for this specific dataset. The architecture consists of two layers: a convolutional layer employing 9 x 9 kernels, followed by a fully connected layer. Consistent with (Quu al., (023b), 1.2 regularization was applied to the convolutional filters to encourage smooth, center-surround-lig like receptive fields, while L11 regularization was applied to the weights of the fully connected layer to promote sparse feature usage for each output neuron. Hyperparameters were determined through grid search. The search space spanned: number of convolutional channels E {8. 16, 24, 32). 1. L2 regularization coefficient E 0.1, 1.0, 10.0), L1 regularization coefficient (0, 10 3 10 1/16, 0.1]. and learning rate {10-2 3,3 x 10 10 3 x 10 (0-4, 10 1000). -4). These ranges were based on the values used the original work (Qiu et al. 20236). Each model variant was trained for 12 epochs using the Adam optimizer (B1 0.99, B2 0.999) with batch size of 32.
Convolutional Neural Network (CNN):
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Training incorporated early stopping based on performance evaluated held-out validatioon/query dataset.
Standard RBF Gaussian Process (GP): The Radial Basis Function (RBF) GP bascline was im- plemented using the identical meta-adaptation script employed for our main model and ablation studies, omitting the feature extraction components. To ensure comparability, the same random seeds were utilized, acknowledging the sensitivity of non-parametric models to the specific conditioning data. GP hyperparameters (kernel lengthscale and variance) were optimized by maximizing the marginal likelihood. We initialized the lengthscale using the median heuristic and employed a broad prior variance (100) to avoid unduly constraining the lengthscale learning process. The model was trained for 300 epochs using the Adam optimizer with a learning rate of 4 x 10 and betas (31 = 0.99, B2 0.999). Early stopping was triggered based on the correlation coefficient measured on the held-out validation/query set.
Linear-Nonlinear (LN) Model: This baseline models neural responses y based on flattened input images (vectorized size 36 x 32 = 1152) according to the function:
y g (Wxx+ + b)
In this formulation, W denotes the weight matrix mapping the input vector the 86 output neurons, b represents a bias vector, and g is pointwise nonlinearity. We employed the hyperbolic tangent (tanh) function for f. Model parameters (W and b) were learned by minimizing the Mean Squared Error (MSE) loss between the predicted responses ý and the ground truth responses y, utilizing the Adam optimizer. 1.2 regularization was applied to the weight matrix W. Hyperparameters were selected via grid search, optimizing the learning rate (6 values logarithmically spaced within [5 x 10 101-)0 and the L2 regularization coefficient (14 values logarithmically spaced from 10-6 to 10a and 0). The optimal hyperparameter combination was chosen and early stopping triggered based on the highest correlation coefficient achieved on a held-out validation set. The model was trained for 400 epochs with batch size of 20, using Adam with betas (31 = 0.99, 32 0.999).
Ablation 1: Identity Deep Kernel This ablation serves to isolate the contribution of the meta-learned deep feature extractor. It removes the feature extractor entirely, directly applying the task-adaptive linear heads (h₂) to the raw input pixels x. The output features from the linear head are then fed into the task-specific RBF GP layer. The task-adaptive parameters (linear head weights W, and GP hyperparameters i) were optimized for each task using the exact same adaptation procedure, hyperparameters, seeds, and initialization strategies described in Section S.F.3 ensuring consistency with the main 'Informed' model.
Ablation 2: Random Deep Kernel This ablation acts as crucial control for the meta-learning process, testing the benefit of the theory-informed initialization against random one, given the same fixed-extractor adaptation framework. uses the same deep kernel architecture as the 'Informed' model (feature extractor é, task-adaptive linear heads h, RBF GP layer). However, the weights of the feature extractor & were initialized randomly using the same scheme as the meta-learning initialization, but without loading the pre-trained weights. Crucially, mirroring the procedure for the 'Informed' model during task adaptation on the biological data, these randomly initialized feature extractor weights were kept frozen. Only the adak-ide parameters (linear head weights W, and GP hyperparameters gp.s were optimized using the exact same adaptation procedure, hyperparameters, seeds, and initialization strategies described in Section [S.F3] This ensures direct comparison of the utility of the frozen meta-learned features versus frozen random features within our adaptation framework.
S.G.3 RUNTIMES
We did not maintain precise runtime logs during experimentation; the following estimates are based on file timestamps, run on a single NVIDIA 1.4 GPU. The full meta-training process for our proposed model, including intermediate evaluations, required an estimated 3.5-4.5 hours. Generating the results presented for our model in Figure 3a (which involved evaluations across 10 training set sizes, for all 86 neurons) necessitated total estimated runtime between 6 and 12 hours. The RBF GP baseline required total computational times comparable to our full informed model ( 6- hours). While the RBF GP lacks neural network computations during adaptation, its runtime may be influenced by factors such as kernel matrix conditioning. The CNN baseline required approximately 5-10 minutes for final training, not counting the hyperparameter tuning time. Runtimes for parametric
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and nonparametric models used here are not directly comparable. The CNN predicts responses for all 86 neurons in parallel, while the adaptation phase for the GP models treats each neuron as a separate task with ID output. We briefly outline some pathways toward scaling the framework to more neurons and larger datasets. The meta-adaptation process is embarrassingly parallel, scaling linearly with the number of tasks (neurons). Our reported runtimes do not exploit this parallelism. Within each task adaptation, CNN forward/backward passes scale approximately linearly in the support set size N, whereas exact GP marginal log-likelihood (MLL) computations scale as ((N)) time and ( memory because of the Cholesky factorization of the N x N kernel matrix. For applications with large support set sizes (N), standard GP approximations like variational inducing
points are recommended (Titsias 2009 Hensman et al. 2013 Ober et al. 2024).
S.H PROTOTYPE EXTRACTION AND INTERPRETABILITY ANALYSIS
This section provides further details on the method used to extract and analyze prototype images, offering insight into the task- representations learned by the linear heads of our deep kernel Gaussian Process model.
S.H.1 COMPUTATION OF PROTOTYPE IMAGES
For visual clarity, opted use the superscript (i) here denote lask-indices, rather than the standard subscript throughout
The "Prototype Image" interpretation method relies on analyzing how the task-specific linear heads h(0)) (with parameters W,), modify the geometry of the data representation learned by the shared feature extractor, o. We start by defining the pairwise Euclidean distances between input images z and vg (vectorized) in the input space:
||ا, =2|122-
jk
Images are then passed through the shared feature extractor ((-):
6; (x3).
The pairwise distances are computed in this shared feature space:
| Φx ox
Subsequently, for each task i, the features are transformed by the tpec-sp linear head h(();
(8)
W,
The pairwise distances are computed again in this task-specific representation space:
D |4 (
As practical implementation detail, before computing the difference matrix A (1) the distance matrices ((() and jjk normalized (by dividing by their respective maximum absolute values) to ensure they are
comparable scale.
The core idea is to quantify how the task-specific head (*( changes the relative distances between pairs of images compared to the shared feature space. This change captured by the difference matrix A (i). ik
D
jk
jk
This matrix indicates pairs of images whose relative distance decreased (A > 0) or increased
(A 0) due to the task-spcc transformation (+))
To isolate the effect relative to the average change induced by the head for a given image j. we zero-center the difference matrix row-wise (i.e., for each fixed image j):
jk
jk
N
jk
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where N is the total number of images considered in the analysis. For result use N 750.
Next, to relate these distance changes back to the input space, we compute a pixel-wise overlap function, (z)(2), between each pair of images I, and ry. This function measures the similarity of pixel intensities aj (u) and ax(2)) at each pixel location u, using a Gaussian weighting:
Osx(w) = expl (a)(w)-ax(w)) 2
Here, a is a bandwidth hyperparameter controlling the sensitivity to pixel intensity differences, and use 0.01, but the method yielded similar results for other small values of explored
For each image j and each task i, we compute per-image contribution map, C (u). This map represents, for image j. the average pixel-wise overlap with all other images k. weighted by the normalized change in pairwise distance Á jk induced by the task-specific head h((), The normalization ensures that the contribution is scaled relative to the total magnitude of distance changes involving image j:
Σ
10 jk
Osk(n)
(w)
jzK
Σ 1
+
Jk
where is small constant (e.g., 10 8) added for numerical stability.
Finally, the overall prototype image p() (u) for task i is obtained by averaging these per-image contributions across all N images:
(w)
1 N
cf (u).
This resulting image (() can be interpreted as visualizing the average input pattern that leads to the largest relative changes in representation distances specific to task i.
S.H.2 ALTERNATIVE NORMALIZATIONS FOR COMPUTING PROTOTYPES
Though the results presented in this work are derived using the difference between maximum- normalized distances heuristic, an approach more robust extreme distance values would be to normalize Ajs by jk rather than normalizing each D by it's maximum. This would define an alternative A'
(*)
D
( jk
A jk
jk
D(
+
Yet another method would first scalar Procrustes factor of aligning the two distance matrices in the least-squares sense,
aD - = Ejek D/C0) D(1) a = arg migmin| (()) D(
Ejer (D(:))
and then define
(r(s)
((0) ik
o D.
jk =
Our present results are robust to this normalization (Fig. the standard A defined in the section above.
S7
so we report both for reference but use
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Original (Fig
LN Filters
Best NLPD
Worst NLPD
Best NLPD (of GPs)
Worst NLPD (of GPs)
Relative Normalization
Procrustes Normalization
Worst
Best NLPD
Worst NLPD
Prototype images obtained using Relative and Procrustes normalization, alongside the original max normalization used in the paper, and corresponding LN-model filters obtained by fitting on the biological data.
Figure S7:
S.H.3 ADDITIONAL VISUALIZATIONS
Here we extend the visualizations presented in Figures 4a and 4c.
Extended 4a Synthetic Data Prototypes: Figure S8 presents additional examples comparing the extracted prototype images (u) with the ground truth linear filters for the synthetic dataset experiments.
Extended 4c Biological Neuron Prototypes: Figure S9
displays prototype images p0) (u) for more extensive set of biological neurons beyond the best and worst 25 examples shown in Figure 4c.
Prototype Images
Linear Filters
d
Figure S8: Additional examples comparing extracted prototypes p(i to ground truth linear filters for synthetic data.
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Better NLPD
Worse NLPD
Figure S9: Prototype images (l) for broader selection of biological neurons, ordered by
performance (NLPD) showing trends consistent with those of figure 4c
S.I BAYESIAN MODEL COMPARISON AND OPTIMALITY VALIDATION
S.1.1 ABLATION STUDY ON BAYESIAN MODEL COMPARISON
To confirm that our method is specifically sensitive the theory's 's structure, we applied our ablation conditions our Theory Informed model for the Bayesian Model comparison analysis. Here, we report the Pearson correlation coefficients obtained by implementing the exact same "identification of optimality level" protocol using the "Random", "Identity". and "Task Ablation" conditions.
Informed Deep
Task Ablation
Random Ablation
Identity Ablation
Kernel
Nveronn Corr 3.60
Peumas Com 0.30
[Paance Cor: 0.32
Terrss Comn eat
Infored Optimality Lovel
Infeceed Oytimaling anty Level
Inferred Optina alty Level
Optimalltty
Optimality Level
Optimality Lavel
Opüimillyy Level
Optimaliy level
Figure S10: Ablation Study on Bayesian Model Comparison shows Theory Informed Features indispensable
When using the unstructured prior, the inferred optimality level is either anti-correlated (Pearson corr. -0.32, Pearson corr. =-0.33 ) with ground truth, or simply not sensitive at all (Figure (STO). This confirms that the model comparison capability is driven specifically by the theory/data match.
S.1.2 MIXTURE KERNEL FORMULATION
To smoothly interpolate between a full explanation by the normative theory and null hypothesis, define mixture kernel K3 convex combination of the theory-informed kernel KTIK and
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standard Radial Basis Function (RBF) kernel KRBE:
K3 K3(z,2') (') 3K (x', x) + (1- 3)K((' ())BF))
Here, E [0, 1] is a mixing coefficient that represents the degree of belief allocated to the theory- informed model. KTIK encodes the prior assumptions derived from the specific normative theory being tested (e.g., efficient coding leading to center-surround receptive fields). The RBF kernel, defined as KRBF(Z, Bg exp null model.
with outputscale fg and lengthscale E, serves as the 1|==x'
21
Our rationale for using the RBF kernel as the null model is that it represents a flexible, general- purpose prior over smooth functions (Williams & Rasmussen 2006), and that the Deep Kernel model itself is an RBF GP operating on top learned feature space. The RBF makes minimal structural assumptions about the data-generating process beyond smoothness, controlled by its hyperparameters (0o, E). Therefore, it provides a reasonable baseline against which the specific structural assumptions encoded in Кπı can be compared. A higher inferred 3 indicates that the specific structure imposed by the normative theory provides a better explanation for the data than the generic smoothness assumption of the RBF kernel.
S.I.3 INFERENCE OF OPTIMALITY LEVEL (3*)
We infer the optimal mixing coefficient 9 by maximizing the marginal likelihood of the Gaussian Process model using the mixture kernel Kg. The marginal likelihood P(Y 3) represents the probability of observing the neural responses Y given the stimuli X and the specific mixed prior defined by B. The optimal 3* found by:
8r =rg mAx P(Y 0,10
This maximization is performed by evaluating the marginal likelihood across discretized grid of B values. Specifically, we evaluated P (X,, for 100 evenly spaced values ranging from 1.
Crucially, before evaluating the marginal likelihood across the 3 grid, other hyperparameters of the model are optimized and then frozen. This includes:
1. Task Adaptive parameters of the informed kernel KTIK, (i.e. linear head weights and GP layer hyperparameters)
2. Hyperparameters the RBF kernel KREF (i.e. outputscale f, lengthscale f, and noise).
These parameters are learned by maximizing the marginal likelihood of their respective base models Once optimized, these parameters are held fixed during the grid search for 8 This ensures that the comparison focuses specifically on the contribution of the informed structure versus the null structure, as interpolated by / (Miynarski et al. 2021).
S.1.4 SYNTHETIC VALIDATION DETAILS
To validate our Bayesian model comparison approach for inferring optimality, we first applied it to synthetic data where the "ground truth" level of optimality could be controlled and measured independently.
Generating Controlled Optimality Receptive Fields (RFs) We heuristically generated synthetic receptive fields (RFs) with varying degrees of optimality using a controlled noise-injection process. The procedure involved the following steps:
1. Initial Optimal RFs: We started with a set of 20 archetypal synthetic "optimal" RFs.
2. Generating " "Anti-Optimal" Noise: To perturb these RFs away from optimality in structured manner, we generated a basis for noise that is explicitly orthogonal to any possible center-surround receptive fields like ours. This was achieved by:
Taking larger reference set of 4000 theoretically "optimal" center-surround RFs (distinct from the initial 20 RFs and any meta- eeta-training data). Let this set be J JRr e R 1000 H
We also considered the transposed versions J
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Flattening these RFs into vectors Jua R
Performing Singular Value Decomposition (SVD) J Jat USVI.
Determining the effective rank of the optimal RF subspace by thresholding the singular values S. We used threshold ratio of 0.1 relative to the maximum singular value.
Identifying the subspace orthogonal to the dominant principal components (columns of V). The projection matrix onto the optimal subspace is P V.V²", where V, contains the first columns of V. The projection onto the orthogonal subspace is Pi P.
Generating noise images by projecting natural images (Nimg) onto this orthogonal subspace: Noiseing Pi Ning- These images represent patterns or structures that are explicitly anti optimal. Figure SII shows examples of such generated noise components.
Example "Noise" images used for generating suboptimal RFs. These images were generated by finding the component of natural images orthogonal to the subspace spanned by large set of optimal senter-surround receptive fields.
Figure S11:
3. Perturbation Process: We simulated random walk for each initial optimal RF over T timesteps (Here, T=600). At each timestep t:
We sample a random "anti-optimal" noise image.
This noise image was scaled by random Gaussian scalar ~ N(0,, scale), where scale 0.01 for our purposes.
The scaled noise was added to the RF from timestep 1-1: RE; = RF + Noise
The resulting RF, was normalized by subtracting its mean and dividing by its 1.2 norm:
RF, (RF, -F.-RF.... RFF) 2| This process gradually pushes the RF away from its initial optimal structure.
4. Subsampling Trajectories: The full perturbation process generates long trajectories (T=600 steps) for each initial RF. To create dataset with distinct, relatively stable levels of subopti- mality, we subsampled these trajectories. For each RF's trajectory of ground truth R² values (Sec. S.I.4). selected smaller number of points (20). The selection aimed capture points along a smooth, approximately linear decay in R²² by minimizing the residual from global linear fit within predefined index windows. This ensures the final synthetic dataset contains RFs spanning range of controlled optimality levels, while avoiding pathological points, for instance when the DoG model fails fit and produces nonsense R² value.
Ground Truth Optimality Quantification: R² of DoG To obtain independent, "ground truth" measure of optimality for the generated synthetic RFs, we used the standard approach of fitting a Difference of Gaussians (DoG) model to each RF and caleulating the coefficient of determination (R²). This motivated by the theoretical understanding that optimal RFs under certain efficient coding assumptions often resemble DoG functions Atick & Redlich 1992).
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Published as a conference paper at ICLR 2026
1. DoG Model: The DoG model is defined the difference of two concentrie Gaussian functions with different amplitudes and standard deviations:
DoG(2, v/0) A. cxp (- (x zo) + (y yo)2
(x - oo)² + (y yo0²
A, exp
2
20²
The parameter vector is 0 = {Ac, As, To, oe J. representing the amplitudes of the center and surround Gaussians, their common center (20, yo), and their standard deviations (o.,
2. Fitting Procedure: For each synthetic RF (dimensions D1 D2), we fitted the DoG model by minimizing the mean squared error between the flattened RF pixel values (RF)) (RFfjla) and the DoG model output evaluated on corresponding grid (DoG (ta(()))
i arg min
|RF flat
DoG (0)]22
0
D1 D2
3. Initialization: Parameter initialization significantly impacts DoG fitting. We used the following strategy:
The scale of the center, Ag was initialized to the maximum value of the RF
The scale of the Surround, A, was initialized to 0.5 x Ac.
log de and log a, were initialized to fixed values corresponding to de se 3.0 and
0g Rd 6.0.
The center location (To- Ho) was initialized using a localized center-of-mass heuristic. This involved calculating ID summed projections of the RF along rows and columns. A ID convolution with a sliding window was applied to these projections to find the region of maximal energy. The center of mass was then computed within this window. To improve robustness, this process was repeated for multiple window sizes The DoG fit was performed for each resulting (æo. yo) initialization, and the fit yielding the highest final Rt was selected.
4. R Calculation: After fitting, the goodness-of -f-fft was quantified using the coefficient of determination (R²):
D1-D2 (RF DoG (0))²
RR -1 DDI-D2 (RF RF
where RF is the mean value of the RF pixels. This Rº value serves as the "ground truth" optimality measure for each synthetic RF, against which our inferred 3 values were compared (in Figure 5a of the main text).
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Published as a conference paper at ICLR 2026
S.J GLOBAL HYPERPARAMETER TABLE
Table S6: Summary of Key Hyperparameters
Parameter Category
Hyperparameter
Value Setting
Optimizer
Adam
Total Epochs (E)
Meta Batch Size (B, num tasks)
10
Total Meta-Trait Tasks
490 (49 batches x tasks)
Meta-Training
Task Data Points (Support Query)
1452
Support/Query Split
5% Support 95% Query (resampled each epoch)
Inner Adaptation Steps (Vinner stepe)
17
Inner LR Linear Head (Orine
x 10 (0.0004)
Inner LR GP Params (OGG)
x 10 (0.002)
Outer Mcta-Update Steps (Muuter steps
Outer LR (Atamee)
x 10-* (0.0004)
Adam Betas (3)
(0.5,0.55
Gradient Clipping (Feature Extractor Norm)
1.0
Initial LR Scaling (First Epoch)
1/10th
GP Noise J,, (Synthetic Tasks)
10-4
Data Type
torch. double
Adapted Components
Linear Heads (h.), GP Hyperparameters (pgp.)
Frozen Components
Feature Extractor (é, both Informed and Random)
Optimizer
Adam 300
Task Adaptation
Epochs
Base Learning Rate (for GP Params)
x 10-3
Linear Head Scale
10 (Effective LR: 10-9)
Adam Betas (8,, 82)
(0.99, 0.999)
L1 Sparsity (Linear Heads)
10
GP Lengthscale Prior
Geussaanmmean-Mdlian Init, var=0.01)
Lengthscale Init
Median Heuristic
GP Noise d, Init (Biological)
Standard GPy GPyTorch Initialization 10
GP Noise d., Init (Synthetic Test)
Architecture
Qiu et al. QQiu et al. 2023b) "SI Branch" Adam
Optimizer
Adam Betas (8,, 32)
(0.99, 0.999)
CNN Baseline
Epochs
12
Batch Size
32
Hyperparameter Selection
Grid Search (Val. Correlation)
Grid: Conv Channels
{8, 16, 24, 24, 32)
Grid: Conv L2 Reg
(0, 0.1, 1.0, 10.0)
Grid: FC L1 Reg
[0. 10-3, 10- 1/16, 0.1)
Grid: Learning Rate
[10-2.3x 10 10 10 10
Adapted Components
GP Hyperparameters (0pp.)
Optimizer
Adam 300 (with Early Stopping on Val. Correlation) x 10
Epochs
RBF GP Bascline
Learning Rate
Adam Betas (51, 32)
(0.99, 0.999)
Lengthscale Init
Median Heuristic
Lengthscale Prior Variance
100 (Broad)
Non-linearity (g)
tanh Adam
Optimizer
Adam Betas (81,82)
(0.99, 0.999) 400
Epochs
LN Baseline
Batch
20
Hyperparameter Selection
Grid Search (Val. Correlation)
Grid: Learning Rate
values log-spaced 10 10-1
Grid: L2 Reg (Weights W)
14 values log-spaced [10- 10'1 + [0]
S26


Reviews:
[{"review_id": "k9eXR17YVa", "reviewer": "Reviewer_raz1", "summary": "The paper proposes a framework for Bayesian meta-learning that turns predictions from a normative theory into a deep-kernel Gaussian Process. The shared feature extractor is meta-trained on synthetic data from an efficient-coding model of the retina, which is then adapted per task to fit real neural activity. The theory-informed kernel enables Bayesian model comparison via a mixture kernel that estimates each neuron’s degree of alignment with the theory. On retinal ganglion cells, the model has higher predictive accuracy and better uncertainty calibration compared to baselines (RBF-GP, CNN, and LN), and in addition, yields interpretable features.", "questions": "- Could the authors discuss how the proposed framework might extend to other neural systems beyond the retina, especially in cases where normative theories differ from efficient coding or are less well defined?\n\n- How does the method scale with increasing dataset size or when modelling populations of neurons?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- The novelty of this study resides in the development of a Bayesian meta-learing framework that links normative theories to empirical observations.\n\n- The study is technically sound, although I have a few points I would appreciate the authors to address (see weaknesses).\n\n- In general, the manuscript is clearly written.\n\n- The results in this study suggest that the developed approach is a valuable contribution towards integrating normative theories and data-driven modelling.", "weaknesses": "- The experiments are limited to a single dataset. It would be desirable to study how the approach generalises to other problems.\n\n- It is unclear how the approach will scale to the very common scenario in neuroscience of the study of neural circuits.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "w7vsdh9tsB", "reviewer": "Reviewer_YV9C", "summary": "This paper introduces a Bayesian meta-learning framework that automatically converts predictions from normative theories (such as efficient coding in vision) into tractable probabilistic models using adaptive deep kernel Gaussian processes. The approach has three stages: (1) generating synthetic data from a normative theory, (2) meta-learning a theory-informed kernel (TIK) on this synthetic data, and (3) adapting this kernel to real biological data. The authors demonstrate their framework on mouse retinal ganglion cell (RGC) responses to natural images, showing improved prediction accuracy over baselines while providing well-calibrated uncertainty estimates and interpretable representations. A key contribution is enabling rigorous Bayesian model comparison to quantify the degree of theory-match in biological data.\n\nThe main contributions are: (1) a general framework for automatically constructing probabilistic models from \"black-box\" normative theories, (2) demonstration that theory-informed priors improve predictive accuracy and uncertainty quantification on real neural data, (3) interpretable representations via prototype images that reveal what task-specific heads learn, and (4) a principled method for validating competing scientific hypotheses using exact marginal likelihoods.", "questions": "1. Can you provide a systematic analysis of what distinguishes neurons where the model performs well versus poorly? Specifically, are failure cases correlated with any known biological properties (cell type, receptive field location/size, response reliability)? For the neurons with low $\\beta^*$ values (Fig. 5b), is this because they do not follow efficient coding, or is it to do with model limitations? Can you analyze whether failures occur for neurons whose true receptive fields fall outside the range covered by your augmented meta-train set?\n2. How would the SVD approach (mentioned above) compare against the current prototype images? Can you validate that the prototype images capture causally-relevant features, for instance by showing that synthetic images emphasizing these prototypes produce predictable changes in model outputs? Finally, for neurons where prototype images look interpretable (Fig. 4c, left), do they match the receptive fields estimated by traditional methods (e.g., spike-triggered average)?\n3. Can you test the model comparison approach on non-natural stimuli (white noise, gratings, synthetic images) that do not possess natural image statistics? If $\\beta^{\\*}$ remains high for neurons with high $\\beta^{\\*}$ on natural images, this would provide stronger evidence that the theory informed kernel​ captures specific EC principles rather than natural image features. Have you considered alternative null models? For instance, what happens if you compare against the \"Task Ablation\" kernel (meta-learned on PCA prediction tasks) instead of the RBF? Would this better isolate theory-specific structure? Finally, for biological neurons with $\\beta^{\\*}$ near 0, can you determine whether this means the RBF null is actually a good fit, or whether both models fail?\n4. You meta-trained on one instantiation of the Ocko et al. model with fixed hyperparameters. Have you explored meta-training on a family of EC models with varied hyperparameters? Would this improve generalization and/or allow the framework to discover which EC hyperparameters best match biology?\n5. You use fixed task-specific linear heads of a specific dimensionality. How sensitive are results to this dimensionality? How would using non-linear task-specific transformations affect the results? Also, the narrow lengthscale on the kernel seems important. Is there a principled way to set this?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "- Originality: the paper makes several original contributions. First, it bridges normative modeling and data-driven approaches in a principled Bayesian framework, addressing a longstanding challenge in computational neuroscience. While meta-learning and deep kernel GPs exist independently, their combination for learning theory-informed priors over functions is novel. The adaptive nature of the prior, where the Bayesian Occam effect automatically relaxes theoretical structure as data becomes abundant (Fig. 4b), is a novel finding in line with theory.\n- Quality: The experimental validation is thorough and rigorous. The authors provide extensive ablation studies (Fig. S1, Table S1) demonstrating that performance gains specifically arise from meta-learned theory-informed features, not from generic meta-learning or random features. The framework also successfully avoids feature collapse (a known pathology in deep kernel GPs) through careful design choices that are well-documented (S.F.4). The validation on synthetic data with controlled optimality levels (Fig. 5a, showing Pearson correlation of 0.88 with ground truth) nicely demonstrates that the method can accurately infer the degree by which the data matches the theory. I am also pleased that the paper also includes negative results and limitations (the \"worst NLPD\" neurons in Fig. 4c), which increases credibility.\n- Clarity: The paper is generally well-written with clear motivation and strong visual presentation. Figure 1 effectively communicates the three-stage framework, and Figure 2 provides a detailed architectural overview. The mathematical notation is consistent, with the theory-informed kernel clearly specified. The supplementary material is exceptionally comprehensive, providing implementation details, hyperparameters, and additional results that would help reproducibility. I appreciate that the main text is accessible, but maintains technical rigor (with additional information in the supplementals where necessary). \n- Significance: this work provides a principled method for validating and comparing normative theories against biological data, a critical need in computational neuroscience. The framework's generality to any normative model where data is available suggests broad impact. Moreover, the ability to perform exact Bayesian model comparison without manual model specification is a useful methodological advancement. The finding that “theory-informed” models excel across all data regimes, and not just low-data settings, challenges conventional wisdom about kernel methods and supports modern perspectives on inductive biases in deep learning. For the machine learning community, this provides a compelling real-world application of deep kernel GPs.", "weaknesses": "1. Validation of this method is restricted to a single biological system (86 mouse RGCs from one retina), testing one normative model of the efficient coding principle. It remains unclear how well this method can generalize to other principles or normative models, and how reliant it is on the specific architectural choices (e.g. CNN feature extractor). To that end, I would have liked to see the method applied to other recent models such as Cadena et al. (2019), Bashivan et al. (2019), or even vision transformers, which would help demonstrate the generality of this method. Moreover, the claim that the framework is “broadly applicable” would be substantially strengthened by demonstrating successful application to at least one additional domain from Table S2 (e.g. V1 simple cells, where sparse coding predictions could be tested, or auditory cortex). \n2. There is insufficient analysis of when and why the method fails. The paper shows that performance varies substantially across neurons (Fig. 4c, 5b), but provides limited insight into what distinguishes neurons where the method succeeds from those where it fails. The observation that poorly-fit neurons have less interpretable and less consistent prototype images (Fig. 4c-d) is descriptive rather than explanatory. Failures might be explained by (a) neurons that genuinely don't follow efficient coding, (b) limitations in the specific efficient coding model used (Ocko et al. 2018), (c) the meta-train set not covering the relevant part of theory space, or (d) fundamental limitations of the RBF kernel in the learned embedding space. The paper would benefit from a more systematic analysis to explain the failure modes. \n3. I have some concerns with the prototype image analysis (Section 4.5, S.H). First, the normalizing procedure for computing $\\Delta_{jk}^{(i)}$ (which normalizes by maximum values) is problematic. For instance, a single distant pair of data points can dominate the normalization, which may induce spurious values in the other data pairs. I suggest seeing how the analysis changes if normalization is calculated through relative distance changes, i.e. $\\Delta_{jk}^{(i)} = (D_{jk}^{\\phi}-D_{jk}^{(i)}) / D_{jk}^{\\phi}$, or through Procrustes alignment. Second, the pixel-wise comparison assumes features exist at the same spatial locations across images, which is a strong assumption given that the feature extractor includes convolutional layers that should be translation-invariant. The visualization method cannot capture features that the head is sensitive to at different spatial locations. I suggest a SVD-based analysis of the head transformation, where $W_i = U \\Sigma V^{T}$. The right singular vectors indicate the most transformed direction in $\\phi$ space, and for each $v_l$ one can compute the gradient $\\partial v_{l}^{\\top}\\phi_{j} / \\partial x_{j}$ to show which features in the image most influence that direction.\n4. In the section on Bayesian model comparison, high $\\beta^{\\*}$ could mean either (a) the neuron follows efficient coding principles, or (b) the neuron’s function is well-described by features useful for natural images generally, or (c) both. To distinguish between these, one could test the model on non-natural stimuli such as white noise, gratings, or synthetic stimuli, where natural image statistics do not apply but where efficient coding principles might still hold. Second, it is not clear what $\\beta^{*}=0$ means. Is it that the neuron is best described by the RBF null model, or that neither the TIK nor RBF kernel captures it well. \n5. The method extracts linearized receptive fields from the EC autoencoder’s bottleneck via LN model fits. Why is the linear readout the right quantity to meta-learn? Moreover, all 490 meta-train tasks come from a single instantiation of the Ocko et al. model with fixed hyperparameters. The paper acknowledges that “varying these parameters would likely yield different efficient coding solutions” (S.E.1). Would meta-training on a family of EC models with varied hyperparameters improve generalization? \n6. From section S.G.3, I see that meta-training tasks around 4 hours, and adapting this to all 86 neurons at all data sizes takes 6-12 hours on a single GPU. For researchers wanting to apply this to hundreds or thousands of neurons, or to test variations of their normative theory, these costs could be prohibitive. I think this paper would benefit from an analysis of how runtime scales with meta-train set size, number of neurons, and data dimensions, and a discussion of possible optimizations such as amortized inference or GP approximations. \n7. What other ways could you have partitioned the meta-learning process other than a CNN feature extractor and linear readout? Could you consider alternatives, such as task-specific kernels beyond RBF in the head space, meta-learning the kernel type itself, or task-specific transformations? Some discussion of why you made these specific architectural choices would help.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "BeNqZGMSVd", "reviewer": "Reviewer_Xmij", "summary": "In this work the authors present a novel framework to capture normative theoreies as deep kernels for a Gaussian Process by framing the extraction process as meta-learning framework. Based on simulation results generated from a theoretical model, the inductive-bias of the theory is sought to be captured by learning a feature extractor that works across distinct instances of tasks, thereby serving as basis of GP kernel that in turn can be used to model and assess real experimental data. The Bayesian model nature of this approach (via the GP) allows for assessment typically difficult in theoretically-motivated models such as the degree of epistemic uncertainty and fidelity of the theory in capturing the experimental data.", "questions": "Questions:\n* Can this framework be applied to other categories of normative theories such as neural sampling code/theory?\n* Would it be viable to meta-learn the feature extractor on multiple instances of theoretical models (e.g. such as models trained and optimized on a distinct set of natural images, but all still trained using EC objective)?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "Clarity: The work is clearly written with easy to follow structure. All necessary background materials were described suffiicently allowing for readers who may not be deeply familiar with the topic to follow and appreciate the contribution.\n\nQuality: All derivations and mathematical motivations are clearly presented and accurate. \n\n\nOriginality & Significance\nThe paper addresses a critical challenge in computational neuroscience — quantitative and comparable assessment of alternative normative theories. As the authors highlight in introduction, the field has been in great need of a method to combine the inductive bias provided by normative theories while providing data-rooted assessment of the merits (or lack thereof) of theory-based inductive bias in improving model predictions.\nThe theory-informed Gaussian process introduced by the authors provide a straightforward approach  to possibly frame theoretical predictions into a meta learning task, capturing the effective inductive-bias into the meta-learned feature extractor. Unlike traditional theoretically-motivated approaches, this method provides for a concrete approach to turn theoretical-model with plausible simulation into ultimately a Gaussian process model that can be fit and evaluated on experimental data. The work is quite original with a theoretically well-motivated application of deep GP kernels in capturing the inductive bias through the formulation of meta-learning framework.", "weaknesses": "Although the bridge from the inductive-bias into meta-learning framework is elegant, the presented case of efficient coding appears quite limited, and leaves applications to a broader theoretical inductive-bias deserved to fully assess the generalizability of this approach. In particular, for the efficient coding example, it appears that meta learning tasks were all derived from a specific instance of auto-encoder model trained on natural images in accordance to efficient-coding objective. Given this, it appears to be an overstatment to me to say that the meta-learning framework can be said to have captured the inductive-bias of the full notion/theory of efficient coding. Rather, it appears that the Gaussian process feature extractor only effectively captured the very specific instance of an autoencoder, and thus not necessary can be deemed to be representative of the underlying theoretical framework.\nIt is also not clear", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "VyNsVumKzl", "reviewer": "Reviewer_SKqL", "summary": "This research presents a new Bayesian machine learning framework to bridge the gap between theoretical neuroscience and data analysis. The method automatically translates complex “normative” theories (like efficient coding) into practical, probabilistic models that can be directly used to analyze biological data. By using a “theory-informed kernel” within a deep Gaussian process, the framework improves prediction accuracy, provides reliable uncertainty estimates, and allows for rigorous testing of the underlying theory – all without requiring manual tuning or specific parameterizations of the theory itself.\n\nThe work delivers a scalable and automated way to integrate theoretical knowledge into data-driven neuroscience, demonstrating improved performance on retinal ganglion cell data. Beyond neuroscience, it also advances Bayesian machine learning by providing a novel application of deep kernels for domain-informed model design, addressing common challenges like “feature collapse” to ensure well-calibrated uncertainty. This framework ultimately allows researchers to not only use theories to understand data, but also test and refine those theories with data in a statistically rigorous manner.", "questions": "Content:\n\n* The authors should elaborate on why it is necessary to have \"task-specific\"\n  linear layers feeding into task-specific GP models. The text introduces the\n  fact but doesn't actually seem to motivate it.\n* Line 266: The authors could elaborate on how the \"base RBF GP\" works when the\n  feature extractor is not used, i.e. how are the input images processed and\n  fed to the linear head and GP in this case?\n* Fig. 3b, Sec. 4.4.: the authors should elaborate on the principal components:\n  how many PCs are needed to faithfully explain the data / why is it adequate\n  to show the GP prediction as function of just PC1?\n* Fig. 3b, Sec. 4.4.: from the plots it seems that the DKL with RBF base kernel\n  has similar behavior to the \"bare\" RBF GP baseline - in both cases the RBF's\n  stiffness / smoothness / low curvature seems to dominate and the model\n  doesn't actually seem to model the data particularly well (esp. in the 1400\n  points case) -- the \"informed\" kernel's effect seems minor.\n* Fig. 5a: The authors should explain the tight cluster of points parallel to\n  the x-axis, they don't seem to be part of the linear trend point cloud.\n* The manuscript introduces \"support\" and \"query\" data (line 140) and equate\n  those to more familiar \"train\" and \"validation\" data, which would make the\n  former two terms unnecessary. Please explain why this new terminology is\n  needed.\n* In several places the authors mention \"well-calibrated (epistemic)\n  uncertainty\". Around lines 1446 and 1589 it is mentioned that the\n  likelihood's noise level is set close to zero. Typically in regression\n  settings, GPs are calibrated when the total uncertainty, in the simplest case\n  of a scalar homoscedastic $\\sigma_n$, $\\Sigma + \\sigma_n^2\\,I$, is\n  considered. If the likelihood noise is (assumed to be) zero, then the authors\n  seem to suggest that the variance in the data is explained by model\n  uncertainty alone. In this case, calibration plots or calibration error\n  metrics (see e.g. http://arxiv.org/abs/2109.10254) should be reported to back\n  the claim that uncertainties are indeed \"well-calibrated\".\n\n- line 265: “the improved performance over the bare RBF GP indicates that the learned embedding is meaningful” such indirect assessments suggest to be glossing over details as the performance of the whole system is taken as an indicator for the performance of a subsystem. Please be more precise.\n\n- line 271: “this subverts the conventional idea about kernel methods, such as GPs, being specialized for extremely low data regimes.” This reads like an overly general statement without references. I relate this statement to the approach documented in the paper, which cannot give rise for a general statement about GPs.\n\n- line 288: “with both models fit on the full dataset” I hope that the models have only been fitted on the training set and a hold-out set or cross-validation was performed. Please reconsider this formulation.\n\n- line 296: “this suggests … our representation usefully structures the input space” no evidence is given for this assessment, please consider removing it.\n\n- line 299: the introduction of the main metric NLPD is very brief. The text does hardly detail where the probability distribution in question is obtained from. Please add details on how NLPD is computed in your use case. This provides a better angle for people in the ML community unfamiliar with this metric to judge your results.\n\n- line 323: 'As N grows, the GP layers ... letting the data \"speak for themselves\".' Please remove this heuristic description of an effect for which the paper does not present evidence.\n\n- page 7, fig 4.a. Please add a color scale, so that the different tones of red/blue can be interpreted\n\nNotation:\n\n* Table S1: please use scientific notation $A \\times 10^{-B}$ vs. ``Ae-B``\n* The manuscript shows inconsistent notation (or typos) in several places:\n  * task $T$ (line 216) vs. $\\mathcal T$ (line 161)\n  * likelihood noise $\\sigma_n$ (lines 1547, 1575) vs. $\\sigma_\\eta$ (line 127 and others)\n  * $\\theta_\\text{gp}$ (line 124) vs. $\\theta_{gp}$ (1375)\n  * $exp$ (line 1375) vs. $\\exp$\n  * text in subscripts such as $\\mathcal D_{train}$ (line 126) vs. $x_\\text{train}$ (line 300)\n  * $P(...)$ (line 404) vs. P$(...)$ (line 416)\n* There is code-specific notation lacking corresponding typesetting, e.g.\n  ZeroMean() instead of $\\texttt{ZeroMean()}$ or better just \"zero mean\" in\n  this case maybe (line 1375), Scikit-Learn’s NearestNeighbors instead of\n  $\\texttt{NearestNeighbors}$, same with n_clusters=None (Sec. S.D.3)\n* line 404 and 416: although the formula in the letter line for $\\beta^*$ is targetting the grid search, please check if the use of $\\beta$ and $K_\\beta$ is correct. If so, please comment why $K_\\beta$ is used in the latter formula on line 416.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "This work demonstrates considerable merit through a novel and conceptually strong approach to integrating theoretical frameworks with data-driven analysis. The authors’ strategy of embedding inductive bias via the Theory-Informed Kernel, learned from simulations generated by normative theory, is particularly intriguing and offers a compelling pathway toward more biologically plausible modeling.\n\nThe methodological foundation built upon Bayesian machine learning is a significant strength, with the provision of calibrated uncertainty estimates representing a valuable addition to the standard machine learning toolkit. Furthermore, the proposed Bayesian model comparison framework appears to be a promising and well-motivated concept, facilitating rigorous hypothesis testing and quantitative evaluation of theoretical predictions.\n\nFinally, the inclusion of detailed technical considerations regarding Deep Kernel Learning (DKL) training, as outlined in Supplementary Figure S.F.4, constitutes a valuable contribution to the field, offering practical insights and potentially mitigating common pitfalls such as feature collapse as previously reported in the literature. This level of technical detail enhances the reproducibility and broader applicability of the presented methodology.", "weaknesses": "The paper appears to be written mainly by neuroscience domain experts,\nmaking it difficult for readers primarily interested in the ML methodology\nto follow due to the use of domain-specific terminology. For instance, the\nmanuscript frequently employs neuroscience-specific terms without introduction\n-- for example, the use of “bottleneck” in expressions such as “minimizing\nbottleneck activity” (line 215) and “single optimized bottleneck neuron” (line\n237) -- which may not be immediately clear to readers from the ML community.\nSection 4.1 and the corresponding supplementary material are challenging to\ninterpret for non-experts; translating the neuroscience terminology into\nmachine learning concepts requires some effort before the underlying regression\nsetup becomes apparent. The interchangeable use of the terms “task” and “neuron”\nis confusing and should be avoided.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "x3ejEvyWam", "reviewer": "Reviewer_JnnY", "summary": "I'm very sorry, but even after three readings I did not understand this paper; it is simply _way_ out of my area of expertise. I'm going to give it a neutral review: goods all around, with the lowest possible confidence. But this review should be totally discarded.", "questions": "If the paper were totally rewritten in such a way that I could understand it, the would help. But it's hardly an appropriate task for the authors. However, if they want to have impact on neuroscientists (at least neuroscientists such as myself), they're going to have to publish a massively dumbed-down version.", "limitations": "", "rating": 6, "confidence": 1, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "n/a; see above.", "weaknesses": "n/a; see above.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)