Discovering Temporally Compositional Neural Manifolds with Switching Infinite GPFA
Authors: Changmin Yu
Maneesh Sahani
Máté Lengyel
Venue: ICLR 2025 Spotlight (2025)
Keywords: Computational neuroscience
neural data analysis
Bayesian nonparametrics
latent variable modelling;
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: We propose a fully Bayesian nonparametric extension of GPFA that enables discovery of temporally compositional neural manifolds underlying high-dimensional population neuronal activities.

Abstract: Gaussian Process Factor Analysis (GPFA) is a powerful latent variable model for extracting low-dimensional manifolds underlying population neural activities. However, one limitation of standard GPFA models is that the number of latent factors needs to be pre-specified or selected through heuristic-based processes, and that all factors contribute at all times. We propose the infinite GPFA model, a fully Bayesian non-parametric extension of the classical GPFA by incorporating an Indian Buffet Process (IBP) prior over the factor loading process, such that it is possible to infer a potentially infinite set of latent factors, and the identity of those factors that contribute to neural firings in a compositional manner at \textit{each} time point. Learning and inference in the infinite GPFA model is performed through variational expectation-maximisation, and we additionally propose scalable extensions based on sparse variational Gaussian Process methods. We empirically demonstrate that the infinite GPFA model correctly infers dynamically changing activations of latent factors on a synthetic dataset. By fitting the infinite GPFA model to population activities of hippocampal place cells during spatial tasks with alternating random foraging and spatial memory phases, we identify novel non-trivial and behaviourally meaningful dynamics in the neural encoding process.

Published as a conference paper at ICLR 2025
DISCOVERING TEMPORALLY COMPOSITIONAL NEU- RAL MANIFOLDS WITH SWITCHING INFINITE GPFA
Changmin Yu'; Maneesh Sahani², Máté Lengyel
'Computational and Biological Learning Lab, Department of Engineering, University of Cambridge 2Gatsb Computational Neuroscience Unit, UCL
CCntter for Cognitive Computation, Department of Cognitive Science, Central European University
ABSTRACT
Gaussian Process Factor Analysis (GPFA) is a powerful latent variable model for extracting oow-dimensional manifolds underlying population neural activities. However, one limitation of standard GPFA models is that the number of latent fac- tors needs to be pre-specified or selected through heuristic-based processes, and that all factors contribute at all times. We propose the infinite GPFA model, fully Bayesian non -on-aaamettiic extension of the classical GPFA by incorporating Indian Buffet Process (IBP) prior over the factor loading process, such that it is possible to infer a potentially infinite set of latent factors, and the identity of those factors that contribute to neural firings in compositional manner each time point. Learning and inference in the infinite GPFA model is performed through variational pnpectationmmaiimiaaae tensions based on sparse variational Gaussian Process methods. We empirically demonstrate that the infinite GPFA model correctly infers dynamically changing activations of latent factors on a synthetic dataset. By fitting the infinite GPFA model to population activities of hippocampal place cells during spatial tasks with alternating random foraging and spatial memory phases, we identify novel non- trivial and behaviourally meaningful dynamics in the neural encoding process.
and we additionally propose scalable ex-
INTRODUCTION
The dominant view of neural coding is that information is represented in the activities of populations of neurons, which supports robust computation (Dayan and Abbott 2005). Trajectories in the high- dimensional neural space are often constrained to a din-dimensional "neural manifold" (Churchland al., 2012; Cunningham and Yu 2014). Hence, to achieve comprehensive system-level under- standing of neural mechanisms, it is essential to be exploratory by firstly uncovering the generic latent factors underlying -igh-dimensional population activities before committing behavioural variables. Advanced unsupervised representation learning methods, such as latent vari- able modelling (LVM), have been developed and applied for such neural data analysis (Churchland et al. 2007, Cunningham and Yu, 2014). However, discovering low-dimensional neural manifolds remains challenging task, not only due to the high variability in neuronal firing, but also because neural representations likely reflect the internal states of the animal, which can vary substantially even while experimentally controlled or observed variables are held constant.
fixed set of
A key limitation of existing LVM methods is the necessity for pre-specifying latent dimensions. This is usually performed through model-selection approaches (Doyal 2007). In the absence of prior knowledge of encoded behavioural covariates underlying neural responses, selecting the latent manifold dimensions based on cross- validation approaches lacks interpretability. and the selection is often sensitive with respect the sampling process. Alternative approaches based regularisation methods, such as automatic relevance determination (ARD: Wipf and Nagarajan 2007; Jensen et al.] 2 Gokcen et al. 2024). require maximum likelihood (ML) learning based on marginalisation over all training samples. These methods thus select a single set of latent factors that most likely account for all observations, but not a potentially different set of latent factors for each observation.
Please send any enquiry changmin n. changmin.yu988mm.. com and lengyel (eng. cam.caam...... aa..ac... uk

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Here propose novel. probabilistically principled model that enables simultaneous posterior in- ference over the number of latent factors and the set of activated latent factors pertinent to each observation. Specifically,
develop fully Bayesian nonparametric extension of the Gaussian Process Factor Analysis (GPFA) model (Yu et al. 2008). popular latent variable model for extract- ing latent Gaussian process factors underlying population activities over single trials. The resulting model, infinite GPFA, incorporates stochastic activation of latent factors in the loading process, which is modelled by the Indian Buffet Process (IBP) prior (Ghahramani and Griffiths 2005). The IBP defines a distribution over binary matrices with a finite number of rows and infinite number of columns, hence enabling inference over the potentially infinite number of features, as well as track- ing uncertainty associated with factor activations for each observation. Unlike existing methods that assume fixed loading process, the infinite GPFA able to infer the temporally dyanmic switching expression of latent factors in each neuron, and this property has important neuroscience impli- cations. As an example, hippocampal place cells exhibit the "dynamic grouping effect", whereby ensemble activities transiently represent spatial locations within different reference frames defined by proximity to corresponding shock zones (Kelemen and Fenton 2010). Despite stationarity in external stimuli, neural spiking frequently exhibits non-trivial temporal structure, potentially due to differential expression of latent behavioural variables in the population activities given changes in internal states of the animal (Flavell et al. 2022).
Learning and inference in infinite GPFA is performed with tractable variational expectation- maximisation (EM). We exploit the sparse variational approach of Titsias (2009), which significantly improves scalability, making it possible to apply the model to real-world datasets. Through empini cal evaluation on synthetic datasets, we show that the infinite GPFA matches the performance of the standard GPFA on dataset with deterministic generative process, but significantly outperforms GPFA when variability is introduced the factor loading process. We apply our model to the popu- lation activity of hippocampal place cells recorded during spatial navigation, and identify non-trivial switching dynamics in the neural encoding process, contingent on the engaged task context.
BACKGROUND
2.1 GAUSSIAN PROCESS FACTOR ANALYSIS
GPFA extends standard factor analysis models, by replacing Gaussian factors with Gaussian Process (GP) factors in order to capture non-trivial temporal dependencies in the latent space (Yu et al. The generative model of GPFA is defined as following (Figure Tap.
2008
fa(:) ~ GP GP (),
for d = -,.....,
(I)
h(x,,) = C. F(xn) +d, y(æn) ~ py (y((a)))((((z)))) for n ......,,
where m.) and (..)) are the mean and kernel functions for the d-th latent factors, respec- tively[] E RMXD is the loading matrix that projects the latent factors to the neural space, with M being the number of neurons, d AM is the offset for the linear transformation, fD(xx,)" is the column-stack stack of all latent factors at input location I,, <(-) is some (non-linear) link function, and 0 represents the set of auxiliary generative parameters.
F(x,)) (xw)
Beyond the simple case with isotropic Gaussian conditional likelihood and a linear link function, learning and inference in GPFA is generally intractable, especially in neuroscience applications where is common to assume an exponential link function and conditional Poisson likelihood. Hence, to deal with such intractability, here we describe the generalised procedures for learning and inference based on variational EM. For scalability purposes, we consider extensions of standard GPFA models with sparse- sparse-variational approximation based inducing points (Titsias) 2009: Adam et al, 2016). Inducing points, U, are function evaluations of latent GPs over selected small number of locations, and can be interpreted approximate sufficient statistics of the latent processes. No- tably, there a simple Gaassian relationship between and F. Letting = [21 represent a complete set of inputs and fx(x) (f()),
IN] [(xN]] the corresponding values of
We assume m" m²(-) unless stated otherwise.
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b
2
Figure I: Graphical demonstration for GPFA and IBP models. Graphical models for generative processes for standard (a) and infinite (b) GPFA models with sparse variational approximation. Illustration of a weighted Gaussian factor analysis model with stochastic binary latent activations.
By taking the limit D co, essentially place IBP prior the binary latent activations, Z
(Equation
the dth latent process, we have
p(fa(x))ua) = = (K (K(z)-'ur Kit K
(2)
where wa denotes the inducing locations for the d-th latent proces 2 and Kd kw
E
RNXS
(x,,, ] n..
We choose the joint variational form over F and U Tiissis 2009). , (F,, U) - p ((fa) with q(uz) = W ST). With Equation 2 these assumptions imply marginal variational approx-
imation for f of the form qffa) = NN 5.).
Hnd 1. (2n, w) (Kiw)) pdg (s ki + kew ((K "'S (Kiw)
(orw))-) 11) ha
The corresponding variational free energy objective takes the form,
Flal Σ (log p(ylo()))))
KL [q(ua))lp(ua)]
Note that q()) is additively GP-distributed (Equation In general, the expected log conditional likelihood can only be evaluated approximately (Duncker and Sahani 2018 Keeley et al. 2020). However, it is possible to compute the expected log
under certain assumptions
of conditional likelihood and link function (e.g., Gaussian observation and identity link function, see Section 3). The KI. divergence between the variational approximation and GP prior over the inducing points can be evaluated analytically. Both variational and generative parameters can be trained via iterative gradient descent such that the free energy is maximised.
2.2 INDIAN BUFFET PROCESS
Standard LVM methods require a pre-set number of latents, which each influence the observations at all times. However, the true number of features is often unknown priori, and the set of active latent factors may vary across different observations. Classical model selection techniques such as cross validation and automatic relevance determination cannot identify observation -ppecfi feature activations. The statistically principled alternative is to perform posterior inference over latent acti- vations given prior distribution. Here, consider the Bayesian nonparametric prior known the Indian Buffet Process (IBP: Ghahramani and Griffiths) (005), which is a distribution over infinite binary matrices that enables simultaneous posterior inference over the number and identity of latent features underlying each observation.
Foor simplicity, we all latent processes share the of S inducing locations, w.
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As motivating example, consider Gaussian factor analysis model with stochastic latent acti- vations (Figure Icp.
fa ~ NN o4);
Wa Beta () C. ~N(,000 for = 1,..,,
D II
p(Z
md
mil Td) TT) N-ma
C(Z, ④ F) on for n = ,,...,N. N.
d=1
where my
Σx=1 ===1
(Znd 1), and represents the Hadamard product.
Given the conjugacy between Beta and Bernoulli distributions, we tractably marginalise out.
D Immma + B)T(N + T(N + +1++8 ))
(ZZ) II
Taking the limit D - oo, the IBP places a prior on [Z], the canonical form of that is permutation- invariant (Ghahramani and Griffiths 2005).
a - ((
(N
-ma)lma 1)!
(4)
II Dn!
N!
d=1
where D is the number of non-zero columns in Z, HN -n=11 is the N-th harmonic number, ma the number one-entries in the d-th column of Z. D, is the number of occurrences of non-zero binary column vector h in Z, is the prior parameter that controls the expected number of features present in each observation.
A useful alternative interpretation of IBP is based the stick-breaking construction (Teh et al. 2007 which defines Ta the product of stick-breaking weights, n. = I Va. where va Beta (r, 1). The stick-breaking construction is mathematically equivalent to the Beta-Bernoulth ap- proximation (Equation B) in the infinite limit, but admits the explicit "sparsity constraint", i.e., the probability of employing the d-th latent factor decreases exponentially with d, and a controls the expected number of latents.
Inference given the IBP prior can be performed with either MCMC or variational methods (Ghahra- mani and Griffiths| 20055 Doshi et al. (2009). Here we briefly review the mean-field variational inference approach given the brekkbreaking formulation of IBP, outlined in Doshi et al. (2009). As- suming factorisation across different latent factors and observations, the variational distributions for the latent variables (and parameters) are defined as following.
q(valaa ba) = Beta(aa, ba), q( (Caa)aa,, Sa) = (Ma,, Sa), for d = 1,..,,..,,D. ((Zna(Tna) (ernoulli(7na),
N, where D is some pre-specified upper bound for the number of latent factors. However, given the sparsity constraint of the IBP prior, the expected number of factors usually much smaller than D. and is controlled by a. It is possible also incorporate a prior over a, facilitating more robust and accurate inference of the effective latent dimension (see Section 3}. Given the conditional independence within the generative model, the variational free energy objective takes the form
for d 1,..., D,n
Flql = (logp(ı, C. log (())))())(()))))
N
Σ (log((TT)) + Σ (logp(C.)) + ΣΣΣ
(log pogp(znan 2nd (T))
(log Z,, C)) + H[gg]
d=1
d=1
1=1d=1
Detailed derivations for the variational objective can be found in Section and Supplemental 1.
Unless necessary, do not explicitly show the variational distributions the expectation is taken with respect for notational simplicity.
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3 INFINITE GPFA
Analogous to the motivation behind the original proposal of IBP, we now propose infinite GPFA, the fully Bayesian nonparametric extension of standard GPFA that allows simultaneous inference over the optimal number of latent features and the set of most likely active latent factors underlying each observation, whilst capturing non-trivial temporal dependencies within expressions of the of instantiated latent factors. Specifically, the generative process of infinite GPFA is the following (Figure lb. details of all derivations this section can be found Supplemental S2):
fa(-) GP
for d = 1, 1...., D,
0d Beta (a, 1), Td =
II % C. C2~,N00,22))
for d I,...,.. D.
a Gamma , 82),
End AT ~ Bernoulli(Ta)
for d = 1, 1,... D D,.,,,..,,N h(x,,) C (Z F(xx)) + d, y(an) p((()=)))((((x.))) for 1,...,, N
(5)
Note that here we leverage the stick-breaking construction of the IBP distribution (Ghahramani and Griffiths 2005). Hence, despite setting an upper bound, the model incorporates sparsity prior allowing automatic model selection of the most likely number of latents. Moreover, as the scaling parameter, a, has a significant effect on the growth of the number of factors, we consider the extended model that integrates over a. To this end, we place Gamma prior over a.
Here we perform variational learning using the finite mean-field variational approximations, (U, Z) ((a) I2_1 (C)))) II, ((na))).
ạ(a (1,E2) (2) = (a quua p.. (H) (u, 1 S
for d 1, 1,..., D. q(Calp N(CClln S)), for d ,,...,,. D. for d = 1...., D. 1...,, Note that so far we have assumed deterministic d, but we could in principle integrate over d with Gaussian prior (see details Supplemental S2).
q(valad ba) Beta( eul ad ba).
((nndTTnd) Tnd) = Bermoull((Tra).
(6)
Given the conditional independence in the generative process, the free energy objective takes the following expression.
Flal
Σ (log Fn. Zn)) KL [x(a))lp(())
ΣKL
KL (o(ua)llpa))))
d=1
D Σ KL (a(va))lp(va)]
- Σ [KL (I(=na)ll((na))))uu
m.d
Most terms in the variational free energy admit analytical expressions apart from the expected con- ditional log-likelihood and the cross-entropy term for the stick-breaking weights, Due to the in- troduction of the binary factor activation matrix, Z, q(h) no longer follows Gaussian distribution, hence previous approximation approaches based on Gaussian quadrature do not apply (Duncker and Sahani 2018). Instead, we leverage Taylor expansion for approximating the expected conditional log-likelihood. which offers effective tradeoff between computational efficiency and approxima- tion 4 Specifically, consider exponential link function and Poisson likelihood, we have the following approximation for the expected log-likelihood term.
M
(logp((yn Z,)))
Σ
Uro (Drm))
exp(hmm) + Var Var[h cxp(hmmm) log ynm!
'It is possible to evaluate the expected log- log-likelihood analytically under the special case of Gaussian condi- tional likelihood with identity link function, see details in Supplemental S2
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where we have replaced (exp hmm)) with its second-order Taylor expansion (see Supplemental S2.2 for details). The variational expectation of h can be easily computed due to linearity, and given the mean-field assumption, the corresponding variance can also be analytically computed using the law of total variance (see expression and derivation details in Supplemental (S2.2).
The cross entropy term in evaluating the expected KL--ivergence with respect to Z is following.
d
(logp(Znd) p(znd) [v))
2nd Σ (log vt) + (1 2nd) log 1
II
2/
(8)
The second term cannot be evaluated analytically. However, we could replace the expected log- product term with a second-level variational lower bound leveraging Jensen's inequality (see Sup-
plemental ST and Doshi et al. (2009) for detailed derivations).
The infinite GPFA model is learned via variational EM, iteratively updating the variational parame- ters (Equation 6). and the generative model parameters (i.e., C. and a), via gradient-based updates that maximise the free energy objective. In practice, we employ a standard automatic differenti- ation library for computing the gradients during training (Paszke et al., 2019). It is also possible to incorporate the IBP prior under the finite Beta-Beun Bernnull approximation (Equation [3). we leave further details in Supplemental [2 We note that due to the fact that we are leveraging variational optimsiation instead of MCMC methods for inference, we cannot effectively infer infinite number of latents, but this lead to minimal impact in practice, see further discussion in Supplemental [S2.4
4 RELATED WORKS
Discovering low-dimensional neural manifolds underlying high-dimensional population activities important for understanding neural computations. There exists various extensions of GP-based descriptive models, such as GPFA and Gaussian Process latent variable model (GPLVM). including
incorporation of pointwise non-linearity (Duncker and Sahani, 2018 Keeley et al. 2020 Zhao and
Park 2017: Wu et al., 2017). full non-linearity based on neural networks (Ashman et al. 2020; Yu
et al. 2022). flexible noisPonsson spike count distributions (Liu and Lengyel| (021), etc. However,
existing methods require model selection techniques for choosing the appropriate latent dimensions, which is largely heuristic and prone to stochasticity arising from cross validation. One notable exception is an extension of GPFA with ARD prior, the Bayesian GPFA (bGPFA; [Jensen et al., 2021 Gokcen et al. 2024), which performs automatic pruning of latent dimensions via MI. learning. However, an important shortcoming of posterior inference with ARD prior is that the most likely set of latent factors selected is based on marignalisation over all observations, hence overlooking potential variability in latent activations across observations. From the scalability perspective, the bGPFA model leverages the circulant approximation for variational GP approximation over all input locations. Such approximation leads to additional error and elevated computational complexity comparing to the sparse variational approximation used in the infinite GPFA model.
(Figure 2c 2eh
Since its introduction, the IBP prior has been tightly associated with factor analysis models (Ghahra- mani and Griffiths, 2005), and has been extended to behaviour modeling (Görür al.] 2006). binary matrix factorisation (Meeds et al., 2006). network link prediction (Battiston et al. 2020), etc. The infinite factorial hidden Markov model (HMM) is a prominent extension that incorporates the IBP prior into generative models with explicit temporal dependency structure (Gael et al., 2008). How- ever, the HMM still admits first-order Markovian and linear latent transitions, whereas incorporating GP latents allows simultaneous modeling of all timesteps, in a non-linear and smoothed manner.
The switching linear dynamical system (SLDS) is a popular class of models for parsing behavioural and neural dynamics into simpler alternating states (Linderman et al., 2017). However, the SLDS assumes categorical switching of latent context variables, whereas the infinite GPFA admits com- positional representation of task context, which is simultaneously more computationally efficient and improves interpretability of latent processes (due to the sparsity constraint given the IBP prior). Moreover, in an SLDS, context switching induces changes only in the latent dynamics, hence requir- ing a neg-negli number of timesteps (depending on the spectral radius of the latent transition operator) for the reflection of context changes in the observation space. In contrast, the compo- sitional nature of factor loading process in the infinite GPFA model allows immediate differential expression of latent processes (that are consistently fitted over all timesteps) into neural activities. Therefore, the infinite GPFA model is more suitable for studying spontaneous neural dynamics in tasks with rapid context-switching structure (Kelemen and Fenton, 2010 Jezek et al. 2011).

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Bayesiun GPSA
0.0
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Figure 2: Empirical valuation of infinite svGPFA on synthetic dataset. a. Generative process for synthetic data, following the standard GPFA generative model with sinusoidal latent processes (f) sin (z) and f2(x)= c3s(3 and (optional) binary masking, Z. The latents are linearly projected to the neural space, and passed through an exponential link function to generate firing Poisson process. b. Variational free energy objective during training for different models. e. R² score between posterior means over latent processes and ground-truth latents (top), and loaded latents (f, bottom), for all models, on data given both generative processes (with (bottom) and without encoding variability). d. Log-log plot of predicted and ground-truth firing rates for SvGPFA (left) and infinite SvGPFA (right). Different color represents different neurons. e. Comparison of reconstruction of ground-truth latent processes (solid lines) based on best linear-fit given posterior means of latent GPs (dashed lines). f. Validating sparseness in inferred latent representations a posteriori based on encoding -urddinnnnnqunnesss metrics (Equation 9) g. Graphical illustration of definition of the uniqueness measures. h. Runtime comparison (average wall-clock time of each EM iteration during training). All evaluations are performed based on averaging over 10 random seeds where applicable.
rates, which are then used to generate spikes following tine-inhomogenene
5
RESULTS
5.1
EMPIRICAL EVALUATION ON SYNTHETIC DATA
We consider a synthetic dataset consists of population spiking generated from two sinusoidal latent processes, following the standard GPFA generative process with exponential link function and Pois- son likelihood (Equation []. To study the effects of variable neural encodings within a single trial, we optionally apply a multiplicative binary mask to the latent processes before projecting them to the neural space (Equation We generate synthetic data for 100 neurons over 10 trials, each lasting 10 seconds in duration, for both cases with and without encoding variability. The data generative process is illustrated in Figure 2a In the latter scenario, hand-design the binary mask such that there are continuous periods when one or both processes are actively encoded into the neural space.
Under both generative processes, fit standard and infinite svGPFA to corresponding population spiking (all hyperparameters are identical for the two models where applicable; values are reported in Supplemental SS). We additionally include the bGPFA model for baseline comparison (Jensen
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et al. 2021). All methods converge quickly under either data generative process (Figure 2b). How- ever, it can be clearly observed that the standard svGPFA converges to a suboptimal free energy value when non-trivial binary masking is introduced, whereas the infinite svGPFA reaches level similar to that achieved with constant latent contributions Despite reaching similar asymptotic free energy objective, we note the practical concern that the bGPFA training is significantly less stable than the infinite GPFA. Upon training completion, to validate the fidelity of fitted latents, we evaluate and compare the R² score between the posterior means over the latent processes and the ground-truth latents (Figure 2c] top panel). For the baseline case with trivial binary masks, we observe that all three models perform comparably well, reaching almost perfect discovery of latent processes driving the generation of neural activities. When stochastic factor loading is introduced to the data generative process, we observe that inferred latents of infinite SvGPFA predicts ground- truth latents significantly more accurately than those of standard svGPFA (one-sided student-t test, p = 0.0028) and PGPFA (p = 2.17 x 10 10-16). Qualitative inspection of the fitted latents suggests that the bGPFA directly encodes the loaded latents (fa(t) = = ft);;; Figure 2e), which we hypothe- sise that this is largely due to the circulant approximation of the full covariance matrix over all input locations (instead of linear smoothing given inducing points). We hence evaluate and compare the R2 score between the fitted latents (Tnd for infinite SvGPFA, where Jedn is the mean of the vari- ational distribution over fa(:) at x.,)) and f, and observe that the fitting between the bGPFA latents and f deteriorates quickly as the latent dimension increases, whilst incurring significantly greater computational complexity (Figure 2h). Comparing to the standard svGPFA model, the decline in model fitting exacerbated in the neural space: infinite SvGPFA predicts the log-firing rates (or cquivalently. ht)) significantly more accurately than standard svGPFA when non-trivial masking is present (Figure 2d). The mean squared error of predicted log-rates is 0.40 + 0.87, which again significantly higher than infinite svGPFA (0.0043 + 0.025). The absence of explicit mechanisms accounting for such variability in standard GPFA leads to failure in learning the correct generative process due to errors induced by periods when at least one of the factors is not activated, such that erroneous gradients is applied to model parameters due to unexpectedly high prediction errors.
The infinite GPFA model is expected to effectively leverage the sparsity constraints induced by the IBP prior and correctly infer the ground-truth number and identity of latent variables. In order to validate this, we set the number of latent processes in the model, D, to be greater than the number of ground-truth latents, Dg. and probe the degree of sparseness within the inferred latent processes. Specifically, we devise the following encoding uniqueness metrics for quantitative evaluation.
H((...))
De EP
D4
(HT(R(
Winf (d)
Ugg-sin((d) =
where
log Dgs
log d
R² (1f.fa)) E. (22
R (o Σ (2
R (mf.r)) Σ R
12a (2( Σj= (1 ..)
R', =
RRi.d =
where H-()) denotes the entropy operator and R² (2 f.) is the R² score between the posterior mean vector of the d-th latent and the i-th ground-truth latent. Intuitively, expected that latent pro- cesses that are most actively loaded into the neural space should encode most information about the underlying processes, and the inferred latents should be encoding only one of the ground-truth latents due to the sparsity constraint induced by the IBP prior. Hence, having sorted the inferred latents based on their averaged posterior responsibilities (Figure 2i bottom), (iiff=gr(d) quantifies how the d-th inferred posterior mean uniquely represent one of the ground-truth latent processes (Figure 2g. blue enclosure), and ugg-sin((d) measures how effectively the top latent processes are representing all ground-truth latents (Figure 2g green enclosure). Hence, if the model has suc- cessfully inferred sparse latent structure with high interpretability. we would expect to observe high uniqueness metric values for latents that are more frequently activated. We indeed observe the ex- pected pattern empirically (Figure 21 middle and top), indicating incorporating the IBP prior helps with identifying the correct set of interpretable latent processes. To further corroborate our claim, we train infinite SvGPFA models with the same number of latent processes as the data generative process, and show that posterior responsibilities of latent processes align well with the ground-truth activation of corresponding latents (Figure Sle).
$See ablation studies for characterising the effects of latent dimensions and expected sparsity level in Sup-
plemental S3.3
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b
roward well
5.6
Cononical Correlations
Free Energy -iggss
0..0
-5.0
SvGPIA infintee SVGPFA bCPRA
66
SVGPFA
SVGPFA
0.5
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200
400 Epochs
600
800
1000
Dimnensios
d
Nomaalsedd Speed
Behavioural Phase
Fato box Vic twititt cate)
50
pln Ar
25
24 hinn (Spetia Map!
time (5)
Figure 3: Probing temporally compositional neural manifolds underlying hippocampal place cell population activities during spatial navigation with alternating behavioural phases. a. I1- lustration of behavioural task (Pfeiffer and Foster (2033). Rats navigate in a 2m x 2m box, with 36 feeding wells uniformly arranged in the box. Animals alternate between searching for reward in a random well (foraging phase), and navigate back to deterministic home well (homing phase). b. Variational free energy comparison (in log-space) of standard and infinite svGPFA, and bGPFA. c. We perform CCA between posterior means over latent processes and selected behavioural variables for the infinite SvGPFA model and baseline models. We show comparison of three principal canon- ical correlations for the two models (dots represent different random seeds). d. Temporal trace of posterior responsibilities associated with selected latent processes, with binary behavioural phase (0 and 1 indicate foraging and homing phases, respectively), and normalised speed at the top. e. Spatial-tuning of posterior means for different masked latents magnitude (fa(æ) = =r(<r)lu$ Red star indicates the home well location. f. Mean square errors of firing rate prediction given infi- nite GPFA versus given only spatial location (p 3.84 10-6; one-sample t-test). All evaluations based on 10 random seeds where applicable.
5.2
TEMPORALLY COMPOSITIONAL NEURAL MANIFOLDS UNDERLYING POPULATION FIRING OF HIPPOCAMPAL PLACE CELLS DURING SPATIAL MEMORY TASKS
Having verified that infinite GPFA correctly infers interpretable latent factors and corresponding ac- tivations for each observation, we now probe the existence of variability in neural encoding and po- tential behavioural implications in real neural data. We apply our model to simultaneously recorded population activities 204 place cells from rat hippocampal CA1, whilst the rat is performing a spa- tial memory task (Pfeiffer and Foster 2013). Within each trial, given 36 uniformly arranged feeding wells within a 2m x 2m open-field arena (Figure Ba). rats Iearn to alternate between foraging for food in an unknown and random location (foraging phase), and returning to a fixed home location (homing phase). Transitions to the next phase or trial is automatic upon consumption of the reward.
We fit infinite svGPFA one recording session lasting 2187 seconds, binning spike trains into spike counts within each 30 ms time window. We instantiate maximally 10 latent processes in the model (see empirical comparison models with varying maximum number of latent factors in Supplemental Figure S2(a)), with 100 inducing points with fixed, equally spaced inducing locations, for each latent factor. Additionally, we standard GPFA and bGPFA to the same dataset for baseline comparisons, using the same hyperparameter setting. Through direct inspection of the free energy comparison, we observe the infinite svGPFA yields greater convergence value than standard svGPFA and bGPFA given the same amount of training, suggesting stronger model fitting (Figure 3b).
To validate the fidelity of learned latent factors, we perform canonical correlation analysis (CCA) between posterior means of latent factors and the experimentally verified behavioural correlates of CA1 place cells, including 2-dimensional allocentric spatial location, speed, and head direction of the animal (0 (e'keeee and Nadel 1978. Geisler et al. 2007 Ormond and O) Keefe 2022). Similar

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to our observation from the evaluations on synthetic dataset (Figure 2c) we found that inferred la- tents from the infinite svGPFA model provide significantly more faithful explanation for relevant behvaioural covariates than those from standard svGPFA and bGPFA, indicated by the high canon- ical correlations over the first three principal directions (Figure 3c additional dimensions yield minimal explained variance of shared covariance given the selected set of behavioural variables, see Supplemental Figure S2(c)). The effective latent dimensions a posteriori is approximately 6 (given qualitative inspection of mean posterior responsibility, see Supplemental Figure S2c), coher- ent with the number of non-trivial principal components within population spiking (Supplemental Figure [S2e). and the dimension of latent manifold underlying CA1 place cells population firing reported in previous studies (Nieh et al. 2021 Yu et al., 2022).
We examine activation probabilities for each latent factor at each timestep. Despite stationarity in the marginal distribution of behavioural variables, we observe high temporal variability in posterior responsibilities for all latent processes (Figure indicating the dynamically compositional nature of low-dimensional neural manifolds underlying population neuronal spikings, By separating the continuous recording into alternating homing and foraging phases, we identify latent processes ex- hibiting selective activations in accordance with different behavioural phases. Specifically, fo(2)) is activated exclusively during the late periods of both homing and foraging phases, and when the speed is close to 0 (see top row of 3d). We cross-check with the spatial tuning of corresponding la- tent factors, computed as the time-average absolute activities of fa(x) = Ta(=)) as a function of spatial location of the rat (Figure Bey. Notably, we observe that fo(=)) is almost exclusively se- lective for the home-well location. The fact that fo(2:) is not only activated upon reaching the home well, but also over the later periods of foraging phase suggests the elevated expression of target lo- cation information of the proceeding trial upon reaching the end of the current trial. Such predictive reactivation of goal location information corresponds to goal-orienter hippocampal replay. which is uncovered here in a purely data-driven manner (Diba and Buzsáki 2007; Pieiffer and Foster 2013 Singer et al. 2013). We additionally examine the spatial tuning for other latent processes, and dis- cover that inferred latents exhibits strong spatial selectivity to environmental boundaries and salient landmarks (e.g., home location). Such observation is coherent with the existing theory of spatial coding with hippocampal place cells that leverages combined information of extended cues in given allocentric directions, specifically with respect to the bounding walls (O'Keete and Burgess 1996).
Despite the discrepancy in spatial tuning between the latents and place cells, the inferred latents nevertheless represent the spatial location of the animal with high accuracy (Figure S20). This hence suggests the possibility that the infinite GPFA discovers faithful and generic neural encod- ings underlying place cells, beyond being merely spatially modulated. We confirm hypothesis through showing that the firing rate prediction based on infinite GPFA posterior predictive distribu- tion is significantly more accurate than predictions based solely neuron-specific spatial ratemaps (p- = 3.84 x 10-6, one-sample t-test; Figure 3t). Our finding coheres with the well known fact that hippocampal neurons exhibit conjunctive selectivity multiple behavioural variables OO'keeee and Nadel 1978 Hardcastle et al. 2017).
6 DISCUSSION
We introduce the infinite GPFA, fully Bayesian nonparametric generalisation of the standard GPFA model. Incorporating the IBP prior over latent activations enables simultaneous inference over both the number of latent factors and the time-van varying activation probabilities of factors. Despite having the intrinsic inconsistency that leads to over-estimation of the number of latent factors (Diaconis) and Freedman 1986 Ghahramani and Griffiths 2005),, we provide extensive empirical evaluations on both synthetic and real neural datasets to demonstrate that the infinite GPFA model is capable of performing accurate automatic model selection and learning of sparse and interpretable latent factors. More importantly, we show that utilities of the infinite GPFA model under the context of exploratory investigations of neural data, which we have identified non-trivial coding mechanisms of hippocampal neurons from a purely data-analytical perspective. Here we have suggested new angle for interpreting neural computations, such that temporal variability within population neuronal activities is attributed to the stochastic loading of latent information, mediated by internal states of the animal (Flavell et al. 2022). Such interpretation could facilitate fundamental understanding of existing experimental data that exhibits changes in neural firings over brief temporal intervals upon context switching, which cannot be explained by existing models that only capture context-
dependent dynamics (Kelemen and Fenton 2010 Jezek al. 2011 Linderman et al. 2017).
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ACKNOWLEDGEMENT
We wish to thank Brad Pfeiffer and David Foster for providing the neural data for analysis in Sec- [.2 We also thank Zilong Ji and Neil Burgess, and anonymous reviewers for helpful discussions and feedback on the manuscript. This work was supported by the Gatsby Charitable Foundation (M.S.) and the Wellcome Trust (Investigator Award in Science 21anee221220 212262/7/18/2 to M.L.).
REFERENCES
V. Adam, J. Hensman, and M. Sahani. Scalable transformed additive signal decomposition by non- conjugate gaussian process inference. In 2016 IEEE 26th international workshop on machine learning for signal processing (MLSP), pages 1-6. IERE, 2016.
M. Ashman, J. So, W. Tebbutt, V. Fortuin, M. Pearce, and R. E. Turner. Sparse gaussian process variational autoencoders. arXiv preprint arXiv:2010. /0/77, 2020.
F. Battiston, G. Cencetti, I. lacopini, V. Latora, M. Lucas, A. Patania, J.-G. Young, and G. Petri. Networks beyond pairwise interactions: Structure and dynamics. Physics reports, 874:1-92, 2020.
D. M. Blei and M. Jordan. Variational methods for the dirichlet process. In Proceedings of the Iwenty-first international conference on Machine learning, page 12, 2004.
M. M. Churchland, M. Byron, M. Sahani, and K. V. Shenoy. Techniques for extracting single- trial activity patterns from large-scale neural recordings. Current opinion in neurobiology, 17(5): 609-618, 2007.
M. M. Churchland, J. P. Cunningham, M. T. Kaufman, J. D. Foster, P. Nuyujukian, S. Ryu, and
K. V. Shenoy. Neural population dynamics during reaching. Nature, 487(7405):51-566 2012.
P. Cunningham and B. M. Yu. Dimensionality reduction for large-scale neural recordings. Nature neuroscience, 17(1) 1 500-1509, 2014.
Dayan and L. F. Abbott. Theoretical neuroscience: computational and mathematical modeling of neural systems. MIT press, 2005.
Diaconis and D. Freedman. On the consistency of bayes estimates. The Annals of Statistics, pages 1-26, 1986.
K. Diba and G. Buzsáki. Forward and reverse hippocampal place-cell sequences during ripples. Nature neuroscience, (2110):241112444 2007.
Doshi, Miller, Van Gael, and Y. W. Teh. Variational inference for the indian buffet process. In Artificial Intelligence and Statistics, pages 137-144. PMLR, 2009.
K. Doya. Bayesian brain:
Probabilistic approaches neural coding. MIT press, 2007.
L. Duncker and M. Sahani. Temporal alignment and latent gaussian process factor inference in population spike trains. Advances in neural information processing systems, 31, 2018.
M. Escobar and West. Bayesian density estimation and inference using mixtures. Journal of the american statistical association, 90(430):577-588, 1995.
W. Flavell, N. Gogolla, M. Lovett-Barron, and M. Zelikowsky. The emergence and influence of internal states. Neuron, 110(16):254-2555 2022.
J. Gael, Y. Teh, and Z. Ghahramani. The infinite factorial hidden markov model. Advances Neural Information Processing Systems, 21, 2008
C. Geisler, D. Robbe, M. Zugaro. A. Sirota, and G. Buzsáki. Hippocampal place cell assemblies speed-controlled oscillators. Proceedings of the National Academy of Sciences, 104(19):8149- 8154, 2007.
11

---

Published as a conference paper at ICLR 2025
Z. Ghahramani and T. Griffiths. Infinite latent feature models and the indian buffet process. Ad- vances neural information processing systems, 18, 2005.
E. Gokcen, A. Jasper, A. Xu, A. Kohn, C. K. Machens, and B. M. Yu. Uncovering motifs of concur- rent signaling across multiple neuronal populations. Advances in Neural Information Processing Systems, 36, 2024.
D. Görür, F. Jäkel, and C. E. Rasmussen. A choice model with infinitely many latent features. In Proceedings of the 23rd international conference on Machine learning, pages 361-368, 2006.
K. Hardcastle, N. Maheswaranathan, S. Ganguli, and L.. M. Giocomo. A multiplexed, heteroge- neous, and adaptive code for navigation in medial entorhinal cortex. Neuron, 94(2):375-387, 2017.
K. Jensen, T.-C. Kao, J. Stone, and G. Hennequin. Scalable bayesian gpfa with automatic relevance determination and discrete noise models. Advances Neural Information Processing Systems, 14:10633-10266 2021.
K. Jezek, E. J. Henriksen, A. Treves, E. I. Moser, and M.-B. Moser. Theta-par peeta-paced flickering between place-cell maps in the hippocampus. Nature, (776))6))2446244 2011.
S. Keeley, D. Zoltowski, Y. Yu, S. Smith, and J. Pillow. Efficient non-conjugate gaussian process factor models for spike count data using polynomial approximations. International conference on machine learming, pages 5177-5186. PMLR, 2020.
E. Kelemen and A. A. Fenton. Dynamic grouping of hippocampal neural activity during cognitive control of two spatial frames. PLoS biology, ((6):e100400) 2010.
D. P. Kingma and J. Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.69800 2014.
Linderman, M. Johnson, Miller, R. Adams, D. Blei, and Paninski. Bayesian learning and inference in recurrent switching linear dynamical systems. In Artificial intelligence and statistics, pages 914 922..2222 PMLR, 2017.
D. Liu and M. Lengyel. A universal probabilistic spike count model reveals ongoing modulations of neural variability. In M. Ranzato, A. Beygelzimer, Y. Dauphin, P. Liang, and J. W. Vaughan, editors, Advances in Neural Information Processing Systems 34, pages 12392-13405 Curran Associates, Inc., 2021.
E. Meeds, Z. Ghahramani, Neal, and S. Roweis. Modeling dyadic data with binary latent factors. Advances in neural information processing systems, 19, 2006.
E. H. Nich, M. Schottdorf, N. W. Freeman, R. J. Low, S. Lewallen, S. A. Koay, L. Pinto, J. L. Gau- thier, C. D. Brody, and D. W. Tank. Geometry of abstract learned knowledge in the hippocampus. Nature, 595(7865):80-88) 2021.
J. Oeeee and N. Burgess. Geometric determinants of the place fields of hippocampal neurons. Nature, 38 (6581)64255-44 1996.
J. O'keefe and L. Nadel. The hippocampus as a cognitive map. Oxford university press, 1978.
J. Ormond and J. O' Keefe. Hippocampal place cells have goal-oriented vector fields during naviga- tion. Nature, 607(7920):441-746 2022.
A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, al. Pytorch: An imperative style, nigh-perffrmance deep learning library. Advances in neural information processing systems, 32, 2019.
B. F. Pfeiffer and D. Foster. Hippocampal place-cell sequences depict future paths to remembered goals. Nature, 49777447::74779, 2013.
A. C. Singer, M. F. Carr, M. P. Karlsson, and L. M. Frank. Hippocampal swr activity predicts correct decisions during the initial learning of an alternation task. Neuron, 1133)1166-11777 2013.
12

---

Published as a conference paper at ICLR 2025
Y. W. Teh, D. Grür, and Z. Ghahramani. Stick-brekking construction for the indian buffet process. In Artificial intelligence and statistics, pages 556-563. PMLR, 2007.
M. Titsias Variational learning of inducing variables in sparse gaussian processes. In Artificial intelligence and statistics, pages 567-574. PMLR, 2009.
D. Wipf and S. Nagarajan. new view of automatic relevance determination. Advances in neural information processing systems, 20, 2007.
A. Wu, N. A. Roy, S. Keeley, and J. W. Pillow. Gaussian process based nonlinear latent structure discovery in multivariate spike train data. Advances neural information processing systems, 30, 2017.
B. M. Yu, factor analysis for low-dimensional single-trial analysis of neural population activity. Advances in neural information processing systems, 21, 2008.
Cunningham, G. Santhanam, S. Ryu, K. V. Shenoy, and M. Sahani. Gaussian-process
C. Yu, Soulat, N. Burgess, and M. Sahani. Structured recognition for generative models with explaining away. Advances in Neural Information Processing Systems, 35:40-53, 2022.
Y. Zhao and I. M. Park. Variational latent gaussian process for recovering single-trial dynamics from population spike trains. Neural computation, 29(5):1293-133 1316, 2017.
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SUPPLEMENTARY INFORMATION FOR SWITCHING INFINITE GPFA
S1 MATHEMATICAL DETAILS OF POSTERIOR INFERENCE WITH INDIA BUFFET PROCESS UNDER THE -RIEA-BR FORMULATION
Here we briefly review posterior inference under the IBP prior with truneated stick boeckbre formu- lation (Doshi et al. 2009). Specifically, we assume the factor analysis generative process, but under the stick- beekkbbe construction
fa ~ O)),
for d = .,..., 00,
C. ~ ~,,, Dd ~ Beta(a, 1),
for d = 1, 00,
for d = 1,.... D
for d = I....,.. D.
(S.1)
2=1
II
nd (1
na) Ta)
Nms
d=1
yn C(Z,, C F) c.
for 1,..., N.
We place mean-field variational distributions the latent variables, v. C, Z. qvaaaas bu) Beta(ad, ba),
Vd.
((Cal (CCalpa, Sa) = (N Sa), Vd.
(S.2)
Vn,d d,
We restate the free energy objective below.
Flu (log C,Z.Y) log 4( ()({))(()))
(ogg((u)))> + Σ (log (C,))(C))
Σ
Σ
(log p( 2nd (u)))
Σ
(log ply, Zn, C)) H][g
(S.3)
The key difficulty in evaluating the free energy lies in the computation of the expected log-density of the prior distribution over Z. Conditioning the stick-breaking weights, the expected log-density of 2nd given v is as following.
log ((za((ZVV))
End v)
Σ Bndd
Σ log e4 +(1 2nd) log
II 2%
(S.4)
The first term in the above equation simple to compute given the Beta variational distribution on V. However, the second term involves the expectation of log (1 (1- v.). which does not
admit closed-form expression. To this end, we employ second-level variational approximation, by leveraging the Jensen's 's inequality.
log Σ qa(l) (1 - 11)) m== 2m
By(w)
log 1 -
II vd i=1
Ey(v)
qa(())
Eg(Waeel)
log(1
vz) + log(um) - log qa(l)
(S.5)
1-1
= Eygl()
(bbb))
+ Σ v(a)) Σ
(am + 6m) + H(ga(()),
m=
m=1
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where (l) is categorical distribution, such that qa(l) qui for all d and L. Note that in the above derivations, we leverage the fact that - - 2 Σl= (1 21) I Um on the first line, and Jensen's s inequality on the second line.
The lower bound shown in Equation S..iiationSSShooldd. for arbitrary 9d, for all We then take the variational derivative of Equation S.5 with respect to ga(t), we could derive the optimal (ā that maximises the lower bound.
La cxp (bbi)
Σ vaa))
Σ amm + bm)
(S.6)
Note that we need to normalise the above equation such that (l) is well-defined.
Substituting Equation S.5 into the computation the log-conditional prior over Z, we can now optimise the resulting lower bound of the variational free energy objective (Equation S.3
S2
VARIANTS OF INFINITE GPFA
S2.1 VARIATIONAL
FOR INFINITE GPFA WITH GAUSSIAN CONDITIONAL LIKELIHOOD UNDER FINITE CETT-BERNOULLI APPROXIMATION
In the main paper, we exclusively work with the IBP prior under the dick-breaking construction. also possible instantiate the IBP prior under the finite Beta-Bernolli approximation. With slight abuse of notations, we term the resulting GPFA model that incorporating the IBP under this construction the "finite GPFA" model. The corresponding generative model is shown following.
fa(-)
GP (0, (.,-))
for d = 1..... D.
(S.7a)
a
Gamma (5,, (2),
(S.7b)
Rd
Beta(a/D D,
for d l.....D. D,
(S.7c)
2nd Ta ~ Bernoulli(Ta)
for d = 1,..., D, n = 1,..,,N
N,
(S.7d)
C.
for d 1,.... D.
(S.7e)
h(xx))
= CZ, O F(xx)) +d,
for n 1,...,
(S.7f)
((x)) p(y(z,)))((((z))))
for
N
(S.7g)
Below we show different variants of the generative process presented in Equation S.7
$2.1.1 DETERMINISTIC LOADING MATRIX AND a PARAMETERS
We start with the simplest case by assuming deterministic E MDD and a E R (i.c., remov- ing Equations S.7e and SS.h from Equation (S.7). Below show the complete derivation of the
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corresponding variational free energy.
Fla
N
N D
(yy FF F.. Zn)))
Σ
KL (g(uu))lp((u))
KL (q(Ta))l((Ta)
Σ
KL (p(2n)))
n=l
d=1
d=1
N
M
Oad + dm) ))
Σ
Uro
20
n1m 172m
d=1
md (Tnalsnal + (w2aa) (s Trn))
KKd
log
ww
N + tr [K - + (1x)) (ka))((mm)) (Kmw) -
(
2
D
I(aa + bb))(())
log
a (ad D ((b(au) - <b(as bal) + (ba 1)(((bb) <sas + bu)) I (aa(aa))(ba)
N
Inn log And + (1 - And) log(1 Tna)l
N
(Tna(1b(aa) - bbas + ba)) + +1 (vb(ba) - clas + ba))]
7=
(S.8)
where (-) is the digamma function. For computing the expected conditional log- ligeliiklhood, we have leveraged the law of total variance identity.
Var [XY]=
(E [x]) Var [Y] + Var [X] (E[)))² + Var [X] Var [Y]
(S.9)
S2.1.2 INCORPORATING PRIOR BELIEF OVER LOADING MATRIX C
We then extend the above model by assuming stochastic loading matrix, C, and infer its value given posterior inference. We place Gaussian priors each row of C, (Equation S.7e and assume the corresponding variational distributions are also Gaussian.
(S.10) Due to mean-field assumption, this lead minimal changes to the free energy objective (c.f. Equa- tion S.8 For the quadratic term in the expected log density, we could readily replace Cmd with E[Ond md (nt) m- The variance component need a bit more workaround, requiring the law of total variance (Equation S.9).
(CC.) = (C,, ( for m=dd
Var Znad Znndnnd nd] [Var (CmdZnadnuc (nnds Snd] + Var [E [Cmd Anddfndd rna) Znnds Snd]
(s )² (Tred ((uha)) + (shr)?) + (nEnd) (Tnalsha)) + Tna(l - Tad) (H
(S.11)
where we have overloaded notations to denote (s )2 (S mm Moreover, there is an additional KL divergence term between the variational and prior distributions over C, which can be evaluated analytically given the Gaussian parametric assumption.
1 Σ
KL (C()))()()) = Σ KL [a(C"))))(C = d=1
M log v log | M (Tr(ST) + (4 (
d=1
(S.12)
Note that so far have assumed deterministic d, but we can also make it stochastic and place Gaussian priors on it. Comparing with the canonical free energy formulation (Equation S.8). this would lead to the inclusion of an additional KL divergence term between the variational Gaussian approximation and Gaussian prior on d, and replace d,, with E[dm], the m- component of mean parameter of the variational distribution over d. The variance component is left unchanged.
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S2.1.3 INCORPORATING PRIOR BELIEF OVER
The scaling parameter of the IBP prior, a, controls the expected number of latent features. Hence, to extend the flexibility of the model, it is preferable to include prior beliefs over a such that it is possible to integrate out a, and simultaneously allow posterior inference over (Escobar and West 1995
Blei and Jordan 2004).
Given the non-negative nature of a, we place Gamma distribution over a, well as assume Gamma variational approximation.
g(alfi, {2) Gamma(al{1.$a),
(S.13)
Assuming stochastic would require replacing terms involving with corresponding expectations with respect to , In the free energy objective, the only terms involve a is KL [[([T)))((() show the updated expression as follows.
I'as + ba)
KL (=))))
Σ 1=1
{log at + log D+ ((aa)T(ba)
ad (a) ((V(as) viaa + ba)) + (ba 1)(((bb)) - <aa + ba)))
(S.14)
Given Equation S.13. we have the following analytical expression for the expectations with respect to a.
()(a)) vi(E)) - log E2, (a)a(a))
(S.15)
E
Additionally, we need include KL divergence term between variational and prior distributions over a, which can also be analytically computed.
KLlglalli.&))lll(asss s2)]
(E1 - s1)((()) log ((E:) log [((1) $1 log
&
E1
$2
1
82
E
(S.16)
S2.2 APPROXIMATION OF EXPECTED POISSON LOG-CONDITIONAL DENSITY WITH EXPONENTIAL LINK FUNCTION
In the main paper, we predominantly leverage the Poisson conditional likelihood and exponential link function. However, the expected conditional likelihood under such parametric assumption given the IBP-distributed binary mask in the loading process does not admit closed-form expression. We hence leverage second-order Taylor approximation for evaluating the variational expectation of the exponential of log-rate under the generative process.
(ogp((y,/FF... Z.)))
Σ Unm (Imm))
(exp hmm))
log yro!!
M
Unm (Nrm)
exp((hmm) + Var Var[ exp(bram)
log Ynm!
m=l
(S.
where we have the expectation and variance of h with respect to the variational distribution as following.
(hmm) Cm (Tn @ (/.) + dm
(S.18)
Var[hnm] C T (s (22 + (s/) T O (1 TT))
where uh and (2 are the mean and diagonal-varte variance of F(x,),, respectively, and @2 represents the elementwise square operation (see Equation S.11.
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S2.3 APPROXIMATION OF EXPECTED GAUSSIAN NOG-CONDITIONAL DENSITY
In the main paper, we show how to leverage second-order Taylor expansion for approximating the expected log-conditional density for Poisson likelihood and exponential link function. Such ap- proximation is sufficiently general, and we can increase the order of approximation at the cost of increasing computational complexity. However, here we show special case for Gaussian like- lihood (with deterministic diagonal covariance matrix) and identity link function where we could analytically evaluate such expectation.
M
1
(logp((ym.F.. Z,)))
20 Σ (yymm hmm))
((ymm (Nmm,))
Var[hmml)
2x²
(S.19)
The expectation and variance of h with respect to the variational distribution can be computed fol- lowing Equation S.18
S2.4 FURTIER NOTE ON THE "INFINITE-NESS" IN INFINITE GPFA
Despite what the name suggests, the infinite GPFA model currently does not support instantiation of an effectively infinite number of latent factors. This is due to the fact that we are leveraging variational inference for fitting model, which requires placing a finite upper bound on the number of factors for tractable inference. However, this leads to minimal negative impacts in practice since intrinsic dimensions of neural manifolds are generally low. Importantly, we wish to emphasise that the main benefit of incorporating the IBP prior is that the Bayesian nonparametric induces the additional flexibility for including new factors only if necessary, and enables temporally varying compositional expression of instantiated factors.
S3 FURTHER DETAILS ON EMPIRICAL EVALUATION
Python implementation of the infinite GPFA model can be access through this repo: https://
github. ccm. changmin www.:///iiwnnnnnnnnneeeaaeaaeeee
S3.1 HYPERPARAMETER SETTINGS
All models are trained with Adam optimiser (Kingma and Bal 2014), with learning rate 0.01. For the main experimental evaluations, we train all models over 2000 epochs. The Bayesian GPFA implementation based on the official Github repository All evaluations are based on averaging over 10 random seeds where applicable.
Synthetic Data. We instantiate both the standard GPFA and infinite GPFA models with stochastic C. where 23 0.1, the infinite GPFA model further places Gamma prior a. with s1 1.0 and #2 = 1.0 (Equation [.13). We set the number of inducing points to be 30 for the main eval- uations, and the corresponding inducing locations randomly initialised and treated learnable parameters. For all models, we use the squared exponential (SE) kernels, with trainable scale and lengthscale parameters.
A (x,,x') = exp exp (-
[||
-|---=11] T
(S.20)
The initial scale and lengthscale parameters are s2 1.0 and 7 0.005 (in time domain) for all models. For all implemented models, we set the latent dimensions, D. to be 10.
Neural Data. We preprocess the spiking train data into spike counts, with 30ms time window. The instantaneous firing rates for each neuron are computed via dividing the spike counts by the time window size, followed by Gaussian smoothing. The loading matrix, C, assumed to be de- terministie, hence is treated as model parameter and is learned through the variational M-step. The
httpp: http:///hh....mmmmcaccccaaaaaagp
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wio variabilty
wi variability
R-aquared Soore
1.0
BYGPFA
Infinite DYGRFF
finite IYGPFF
DvGPFA
intrits-w-GPp
Erite exGPFA
So pradictio)
-0.4
Mean Responability
0.6
Free Energy
-0.8
0.4
Seweegg 21 Sig
-.2
syGPFA
0.2
Frile-svGPFA
finito svGPFA
1.8
nfnitt-svGPFA
infinite evGPFA
500
Epocha
1000
1500
2000
10g rate (Ground 21th)
2 0
0
Soried Latent Dimensions
2
4
6
8
f
1.0
posterior responsibility (Tns)
ground-truth activation
1.0
probability
0.5
AUC
0.5
0.0
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
time (s)
f200
f((
Figure S1: Further empirical evaluations of infinite svGPFA on the synthetic dataset. a. Com- parison of R² score between posterior mean of latent processes and ground-truth latents for all three models (standard svGPFA (blue), infinite svGPFA (orange) and finite svGPFA (green)). b. Free energy comparison between the three models. Log-log plot of predicted firing rates against ground-truth firing rates for fitted finite SvGPFA model. d. Mean responsibility comparison between finite and infinite SyGPFA. e. Temporal trace of posterior responsibility for a latent process and the ground-truth activation for the corresponding latent (up to permutation). f. Area Under Curve (AUC) for binary classification of latent activation given posterior responsibilities.
concentration parameters, a, is again assumed to be stochastic, with Gamma prior and parameters = 1.0, $2 1.0. For all models, the number of inducing points are 100, and corresponding inducing locations are fixed as equally spaced location along the input (time) domain. We use the SE kernels for the latent GPs with trainable scale and lengthscale parameters. For all implemented models, we set the latent dimensions, D, to 20.
Starting and end times for foraging and homing phases are provided as part of the data collected by the experimenter (Piciffer and Foster 2013).
S3.2 FURTHER EMPIRICAL EVALUATION ON SYNTHETIC DATA
In addition the standard and infinite svGPFA with stick-breaking formulation, also implement and evaluate the finite SvGPFA model (under the finite Bernoulllll approximation of the IBP prior) the same synthetic data in the main paper (Figure 2a).
From Figure Sla we observe that the finite svGPFA model yields stronger performance than the standard SvGPFA model, in terms of the R² score of the linear fitting between the posterior mean of
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Figure S2: Further empirical evaluations for fitting infinite SvGPFA to the neural data. First three canonical correlations between the posterior mean of the latent processes and the selected be- havioural variables, for standard svGPFA, and infinite svGPFA with varying numbers of maximally allowed number of latent factors. b. Correlation heatmap between canonical correlates of inferred latents and relevant behavioural variables. c. Mean posterior responsibilities (sorted). (d). Canoni- cal correlations between posterior mean of the latents and the set of selected behavioural variables, over the first 5 principal directions. Explained variance ratios over the top 20 principal directions underlying the population spiking data. g. Ground-trutt (solid) versus predicted (dashed) x- and y-position on heldout dataset.
the latent processes and the ground -ruth latents. We observe slight performance drop compar- ing to the infinite SvGPFA, but such drop is insignificant (p 0.094). Similarly, by inspecting the free energy comparison and the log-log plot between the ground-trui -ruth firing rates and the predicted firing rates given the finite svGPFA model, observe that the finite svGPFA leads to similar model fitting performance (Figure SIb Slc).
The finite SvGPTA, due the finite esta-Bernoulli formulation, all factor probabilities na follows the same Beta distribution (TT ~ Beta ( 1)). Hence, unlike the stick- breaking construction, we do not expect the finite svGPFA instantiate explicit sparseness constraints, hence does not have the capability to perform automatic model selection for the number of latent factors. Indeed, by inspecting the mean posterior responsibilities associated with the latent factors, observe that the
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Figure S3: Ablation Studies Infinite GPFA Synthetic Data. a-c. Temporal trace of free energy during training for standard and infinite svGPFA models, under conditions with and without encoding variability, for varying latent dimensions. d. Number of training epochs to reach conver- gence (d) as function of latent dimensions. e-f. Difference (in percentage) between converged free energy objective values between standard and infinite SvGPFA, for varying latent dimensions (e), and varying expected sparsity level in the loading process (f).
finite svGPFA essentially instantiates all latent processes for (differentially) representing ground- truth latents, whereas the infinite svGPFA effectively leverages the top factors instead of attributing the explanatory power to all latent processes (Figure STd).
To corroborate our analysis in Figure 2 we train infinite svGPFA models with the same number of latent factors the data generative process. After fitting, we compare temporal traces of posterior responsibilities of inferred latents and corresponding ground-truth activations (up to permutation). We observe that the posterior responsibilities are well-aligned with the actual activations, suggest- ing the model has correctly identified the dynamic loading of latent factors onto the neural space (Figure Sle SIf).
S3.3 ABLATION STUDIES OF INFINITE GPFA ON SYNTHETIC DATA
In the main paper, have verified that the infinite GPFA outperforms standard GPFA model when there exists non-trivial multiplicative binary masking in the loading process (Figure 2 However, there exists potential limitation given observations from empirical evaluations: the infinite GPFA models take significantly more training epochs than the standard GPFA models to reach convergence Practical application of the proposed model would become prohibitive if such lag- ging in convergence scales positively with the latent dimension. We validate this by comparing the temporal trace of the training objective between standard and infinite GPFA models, whilst varying the number of latent processes (Figure S3h-c). Specifically, the (maximally) 8 latent processes are as following.
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fr(z) = sin? (2x) cos(r), fs(x) = cos c2x) sin(x),
We quantify the amount of training to reach convergence by computing the number of training epoch to reach 95% of the asymptotic free energy value. Quantitatively, we indeed observe that the infinite GPFA takes more training epochs to reach convergence (Figure 531). However, under the trivial-
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masking condition, the amount of training for the infinite GPFA model to reach convergence does not increase with the number of latent processes. Additionally, under the condition where there non- trivial binary masking, despite there exists a positive correlation between the amount of training to reach convergence and the number of latent processes, the magnitude of the positive correlation for the infinite GPFA model is lower than the standard GPFA model. Importantly, as the number of latent processes increases, the gap in the converged free energy objective (under the condition where there non-trivial binary masking) significantly greater for the standard GPFA model (Figure S3 L Collectively, the convergence rate should not be bottleneck in practical application of the infinite
GPFA model, and the significant performance difference as the latent dimension increases yields the infinite GPFA model as the more favourable option empirically.
We additionally verify the effect of expected sparsity in the binary mask. We define the sparsity as the expected number of 1 (active) entries in the binary mask. Cohering with our expectation, the performance difference between the standard and infinite GPFA (2 (oit peNGPEA the expected sparsity in the mask increases (Figure S3D.
-100) decreases as [
S3.4 FURTHER EMPIRICAL EVALUATION ON NEURAL DATA
Given trained infinite svGPFA model the neural data, we choose the effective latent dimension through inspecting the temporal mean of posterior responsibilities associated with the inferred latent processes (Figure S2ch. By choosing the threshold of 0.05, we identify 6 non-trivially activated latent processes, which is indeed coherent with results reported in literature (Nich et al. 2021 Yu 2022). To further corroborate the discovery, we perform standard cross validation model selection for selecting the optimal number of latent variables. We use the CCA correlations as the evaluation metrics, we observe that model performance saturates when reaches (Figure (S2a). Moreover, through directly performing PCA on the population spiking, we observe that the majority of the variance is explained by its first 6-7 principal components (Figure (S2e). These observations provide additional empirical validity for the correctness of the optimally inferred number of latents given the IBP prior.
In Section 5.2 of the main text, given the selected set of behavioural variables, focus analysis on the first three canonical correlates (Figure [c). This decision can be justified by the empirical observation that there are only three dominant dimensions explaining the covariance between the latent processes and the selected behavioural variables (Figure S2d
We quantify the gap the percentage of decrease in free energy convergence given existence of trivial and non-trivial binary masking, yrovenal riaa
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Reviews:
[{"review_id": "5qRDJY7q8z", "reviewer": "Reviewer_iwdZ", "summary": "This paper proposes the infinite GPFA model, which is Bayesian non-parametric extension of the classic GPFA by combining GPFA with an Indian Buffet Process Prior. This model can potentially infer infinite set of latent factors from data. A variational EM algorithm is proposed to perform the inference. The authors demonstrate the effectiveness of this model through analysis on simulated and real datasets.", "questions": "- unclear sentence line 366: \"Moreover, in an SLDS, only the latent dynamics changes following context switching, hence requiring a non-negligible number of timesteps (depending on the spectral radius of transition operator) for the reflection of context changes in the observation space. in contrast, the compositional nature of factor loading process in the infinite GPFA model allows immediate differential expression of latent processes into neural activities.\"\n\nCan you clarify this a bit? infinite GPFA model seems to also have the factor loading process in latent space, why it allows immediate expression into neural activities than SLDS? \n\n- How is the number of features D selected for svGPFA in the experiments section for synthetic data and real data?\n\n- What's the future research direction for this paper?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The paper is clearly written. The model formulation and related works are clearly introduced. \n- The authors have done extensive experiments on real neural data and synthetic data, and results seem good.", "weaknesses": "- The idea of combining GPFA with IBP prior is not revolutionary. \n- I listed some questions in the section below.", "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": "q4wkbsgUEz", "reviewer": "Reviewer_LfyK", "summary": "The authors present an extension to GPFA, a widely used latent variable model in neuroscience, that uses an Indian Buffet process as a nonparametric extension to automatically select latent dimensions at each time point. This avoids the need for a priori latent dimensionality choice in GPFA, a well-known limitation to the method, and allows for a sparse selection of latent activations at each time point, which can identify transitions in the latent representation, enhancing the models usefulness in the identification of behavioral states. The authors show strong validation on synthetic datasets as well as real spiking data. The theory is clear and model development and implementation is clear and sound.", "questions": "None", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "The switching GPFA and switching infinite GPFA models effectively tackle a significant limitation commonly encountered in many latent variable models in neuroscience, particularly within GPFA: the a priori selection of latent dimensionality. Additionally, these models enhance the approach by allowing for unequal contributions of latent variables at different time points, addressing another critical shortcoming of traditional GPFA. This advancement represents a noteworthy contribution to latent variable modeling in neuroscience. The authors also incorporate inducing points for improved scalability, a practical and well-established extension from the existing GP literature.", "weaknesses": "The weakest part of the manuscript is the lack of evaluations to any competing approach. The authors appear only compare to variants of their own model. In particular, because the authors emphasize the advantage of not needing to pre-select latent dimensionality, some evaluation against the ARD approach in Jensen et al would be appreciated. The authors claim the ARD is inferior due to requiring marginalizing over all of the data to determine latent dimensionality, and this is sound reasoning, however, I am curious as to how exactly different the models fits and latent posteriors would be. It might be possible, for example, for the ARD GPFA model to learn an appropriate number of latent dimensions and have periods of time where different groups of latents are minimally or highly variable. I think it would help a reader get a sense of how svGPFA compares Bayesian GPFA, as the latter is a model that was motivated in a very similar way.\n\nNote also that the manuscript \"Uncovering motifs of concurrent signaling across multiple neuronal populations\", Gokcen et al. also uses an ARD prior in a similar GPFA-style model - might be worth citing\n\nOne small point -- Figure 2 is difficult to render in the browser and this specific page lags. I suspect the figure size is too large, maybe due to panel d. Downsampling this figure before adding it to the latex might 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": "Qhrpl2NiRT", "reviewer": "Reviewer_52sh", "summary": "The authors propose a novel model, an extension to GPFA, that incorporates stochastic activation of latent factors in the loading process via IBP prior. This results in dynamically switching expression of latent factors for each neuron across different timepoints, hence incorporating the dynamic shifts in internal states of the animal. They apply their model, infinite GPFA, to two datasets, one synthetic and one real world  neuroscience dataset (hippocampal place cells during spatial navigation).", "questions": "- How does the infinite GPFA handle cases where it identifies overlapping/slightly differing  latent factors ?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "- **Novelty**: The proposed model, infinite GPFA, has a robust mechanism that allows for  estimation of both the number of latent factors and their time-varying activations without requiring manual tuning. In addition, the sparsity allows for learning of more interpretable latent factors, which is helpful for interpreting neural computations.\n- This framework opens up new avenues in neuroscience for exploratory investigations of experimental data.\n- Presentation is clear.", "weaknesses": "- More comparison to other methods could have strengthened the utility and performance of infinite GPFA, specifically, using some of the previously established methods like GPFA with ARD prior. Although GPFA with ARD prior is not designed to capture latent factors across time, it would be useful to show it quantitatively.\n\nMinor points\n- l060 ‘An as example,’ → ‘As an example,’\n- Figure2.a the axis labels are illegible .\n- In general figure 2 gets rendered very slowly, I am not sure the exact cause but it might be worth investigating because if it’s simple like rasterization or high resolution graphics, it can be easy to fix.", "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": "yWakuFSnGu", "reviewer": "Reviewer_swfp", "summary": "the authors introduce switching infinite gpfa — an extension of the classical gpfa model to account for the possibly time varying dependence of observed neural activity on different latent factors.  the authors make this possible by using an indian buffet process as the generative model for a (infinite) binary mask that selects the latent features read out  to the observation space at each point in time.  they outline how to perform tractable variational EM inference/learning for this model class.  the authors then validate their model and inference/learning procedure on synthetically generated data, and then show how their method can be used to extract behaviorally meaningful latent features from rats performing a spatial navigation task.", "questions": "for fig.2c what expected level of masking for Z was used?  on that topic, it could also be interesting to show how the gap between gpfa and infinite gpfa inference performance varies with the expected value of alpha.  additionally, an inline figure or addition to figure 1 helping to build intuition between numerical values of alpha and the expected number of features could be useful.\n\nis the runtime comparison in Fig.2h for a single EM iteration, or only inference time?", "limitations": "", "rating": 8, "confidence": 5, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "the paper is very well written. the background section is clear and in my opinion and succesfully takes the reader from the original gpfa to their new generative model that incorporates an indian buffet process prior over the binary masking matrix, Z.   since approximate inference in this model is highly non-trivial, the authors developed an approximate variational EM procedure for inference/learning.  i appreciated the extensive discussion covering what terms are and are not tractable in the variational bound and how the authors deal with the intractable terms in a practical manner;  important details that would clutter the main text were referenced often and helped with further clarifications. their synthetic data example validates their inference approach and reassuringly shows the infinite gpfa model can match standard gpfa inference quality even when there is no encoding variability.  in their last experiment, they apply their method to neurophysiological recordings taken from a rat performing a spatial navigation task;  they demonstrate how their method can reveal the compositional structure of the latent space by identifying different latent factors being loaded onto the neural population at different locations or trial events.", "weaknesses": "more comparisons could be helpful.  for example, it could have been interesting to see how bGPFA also compares to infinite gpfa with and without model mismatch similar to the synthetic example presented.  \n\nfrom fig 2b and fig 3b, it does appear that infinite gpfa takes substantially longer to reach convergence.  do the authors expect this difference gets substantially worse with higher latent state dimensionality? it could be helpful to see convergence plots for a dataset that requires higher latent state dimensionalities.", "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 (Spotlight)