Multiway Multislice PHATE: Visualizing Hidden Dynamics of RNNs through Training
Authors: Jiancheng Xie
Lou C Voinov
Noga Mudrik
Gal Mishne
Adam Shabti Charles
Venue: Accepted by TMLR ()

Abstract: Recurrent neural networks (RNNs) are a widely used tool for sequential data analysis; however, they are still often seen as black boxes. Visualizing the internal dynamics of RNNs is a critical step toward understanding their functional principles and developing better architectures and optimization strategies. Prior studies typically emphasize network representations only after training, overlooking how those representations evolve during learning. Here, we present Multiway Multislice PHATE (MM-PHATE), a graph-based embedding method for visualizing the evolution of RNN hidden states across the multiple dimensions spanned by RNNs: time, training epoch, and units. Across controlled synthetic benchmarks and real RNN applications, MM-PHATE preserves hidden-representation community structure among units and reveals training-phase changes in representation geometry. In controlled synthetic systems spanning multiple bifurcation families and smooth state-space warps, MM-PHATE recovers qualitative dynamical progression while distinguishing family-level differences. In task-trained RNNs, the embedding identifies information-processing and compression-related phases during training, and time-resolved geometric and entropy-based summaries align with linear probes, time-step ablations, and label--state mutual information. These results show that MM-PHATE provides an intuitive and comprehensive way to inspect RNN hidden dynamics across training and to better understand how model architecture and learning dynamics relate to performance.

Published in Transactions on Machine Learning Research (02/2026)
Multiway Multislice PHATE: Visualizing Hidden Dynamics of RNNs through Training
Jiancheng Xie Deportment of Biomedical Engineering Boston Universily
ziejo edu
Lou C. Kohler-Voe Voinov
Kohnor-Voinov
low.kohlerronnoos@@pp.....
Department Life Sciences Ecole Polytechnique Fédérale de Lausanne
Noga Mudrik
umudrik1 @ihu edu
Department of Biomedical Engineering, Kavli ND1, Center for Imaging Science
Johns Hopkins University
Gal Mishne
gmishne Gucad. edu
Hakcroglu Data Science Institute Universily of California San Diego
Adam S. Charles
adamse jihau edu Department of Biomedical Engineering, The Mathematical Institude for Data Science, Center for Imaging Science
Johns Hopkins University
Reviewed OpenReview: /ttps://openreveee
Ret/ for/formm? id= 9Yr4V722gg
Abstract
Recurrent neural networks (RNNs) are widely used tool for sequential data analysis; how- ever, they are still often seen as black boxes. Visualizing the internal dynamics of RNNs a critical step toward understanding their functional principles and developing better architectures and optimization strategies. Prior studies typically emphasize nelwork repre- sentations only after training, overlooking how those representations evolve during learning. Here, present Multiway Multislice PHATE (MM-PHATE), graph-based embedding method for visualizing the evolution of RNN hidden states across the multiple dimensions spanned by RNNs: time, training epoch, and units. Across controlled synthetic bench- marks and real RNN applications, MM-PHATE preserves hiddeen-representaaiin community structure among units and reveals training-phase changes in representation geometry. In controlled synthetic systems spanning multiple bifurcation families and smooth stale-space warps, MM-PHATE recovers qualitative dynamical progression while distinguishing family- level differences. In task-trained RNNs, the embedding identifies information- pttpraattiioo-ooessinnnn and compression-related phases during training, and timo-resolved geometric and entropy- based summaries align with linear probes, time-step ablations, and label-state mutual informa- tion. These results show that MM-PHATE provides an intuitive and comprehensive way to inspect RNN hidden dynamics across training and to better understand how model archi- tecture and learning dynamics relate to performance.
1 Introduction
Recurrent neural networks (RNNs) are designed handle sequential data by modeling input sequences and retaining memory of past elements through recurrent connections or memory units (Lipton et al. (2015] Kaur & Mohta 2019). Unlike feedforward neural networks (FNNs), which process each input independently, RNNs

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update their internal state dynamically, enabling them to capture contextual relationships across sequences. This makes RNNs particularly effective for tasks that rely on the order and relationships between elements, such as time-series analysis and action recognition (Hewamalage et al. 2021).
Since gaining popularity in the 1990s, various RNN variants and training strategies have been developed to improve training stability and long-range dependency learning. Architectures such as Long Short-Term Memory (LSTM) networks (Hochreiter & Schmidhuber) [996) and Structurally Constrained Recurrent Net- works (SCRN) (Mikolov et al.) 2014) were designed to address challenges like the vanishing gradient prob- lem @emlShehin et al. 2018). Additionally, RNNs excel at handling irregular or incomplete sequential data due to their fiexibility with variable-length inputs. These properties, along with ongoing architectural im- provements (Salehinejad et al.) 2018), have enabled RNNs to achieve exceptional performance across domains such as natural language processing (NLP), neuroscience, and biomedical signal processing (Barak 2017] Chen & Li 2021 Khalifa et al. 2021). Notably, RNNNs remain the state-of-the-ar for neural decoding intracortical brain-computer interfaces (Deo et al. 2024).
Despite their extensive use, the learning dynamics and internal representations of RNNs remain difficult to interpret. This opacity complicates performance evaluation, model design, and parameter selection, hindering the development of more effective architectures. While significant progress has been made in interpreting FNNs (Wang et al. 2021 Yu et al. 2014), similar advances for R.NNs have been limited. Most RNN analyses focus on either fixed-point analysis in dynamical systems (Sussillo & Barak, 2013 or performance evaluations across different architectures and components (Chung et al.) 2014 Greff et al. 2017). Consequently, there is a clear need for new methods that facilitate the interpretation of RNNs' latent representations during training.
In explainable deep learning, dimensionality reduction is widely used to visualize high-dimensional data, helping researchers gain intuition about its structure and relationships (Karpathy et al. 2015 Hidaka & Kurita, 2017: Rauber et al. 2016 Gigante et al., 2019). However, existing techniques often have limita- tions. Methods like t-SNE emphasize local structure at the expense of global patterns (Maaten & Hinton 2008), while methods such as PCA and Isomap focus on global structures and may miss important local de- tails (Makkiewicz & Ratajezak 1993 Tenenbaum et al.) 2000). These methods can also be sensitive to noise and outliers (Moon et al.) 2019) Visualizing RNNs is particularly challenging since they require prescrving structures across nultiple dimensions, including time, epochs, and hidden units. Consequently, traditional dimensionality reduction techniques often fail to capture the complexity of RNN learning dynamics.
To address these challenges, Gigante et al. GGigante ct al. 2019) introduced Multislice PHATE (M-PHATE), which visualizes FNN hidden states during training. M-PHATE constructs a multislice graph where each slice represents the network's state at a particular training epoch, capturing both inter-epoch relationships and community structures through PHATE(Moon et al., 2019). This approach effectively captures performance- related features, such as task-related specialization, without requiring external validation data, making it particularly useful in data-limited settings.
However, M-PHATE is designed for FNNs and does not account for the sequential nature of RNNs, where hidden states across all time-steps play a critical role in representation learning (Su & Shlizerman 2020). To address this limitation, we propose Multiway Multislice PHATE (MM-PHATE), which extends M-PHATE by capturing RNN hidden states across both time-steps and epochs. MM-PHATE provides a comprehensive view of RNN learning dynamics, enabling to track how hidden representations evolve during training and how they support task performance. These training dynamics are reminiscent of ideas from information bottlencck theory, where intermediate representations are encouraged to retain task-relevant information about the target while discarding irrelevant variability in the input (Fischer 2020 Tishby & Zaslavsky 2015 Cheng et al. 2019). In contrast to classical information bottleneck approaches, we do not optimize or estimate a formal bottleneck objective; instead, we empirically characterize "expansion" " and "compression" phases via the geometry and entropy of hidden-state trajectorics revealed by MM-PHATE. We evaluate MM-PHATE on controlled synthetic benchmarks and real RNN applications, including synthetic systems spanning multiple bifurcation families and smooth state-space warps, and compare against existing methods such as PCA (Wold et al. 1987). L-SNE (Maaten & Hinton 2008). Isomap (Tenenbaum et al.) 2000), Locally Mclnnees et al. 2020), and M-PHATE.
Linear Embedding (LLE) [Roweis & Saul 2000), UMAP
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Our main contributions are as follows:
We introduce MM-PHATE, multiway multislice visualization framework for RNN hidden dynamics across time-steps and training epochs, providing unified view of representation evolution during learning.
We evaluate MM-PHATE on controlled synthetic benchmarks (Hopf and Pitchfork bifurcations, with and without smooth state-space warps), showing that it recovers qualitative dynamical progression and preserves family-level distinctions under the tested conditions.
We show on fask-trained RNNs that MM-PHATE preserves hidden-unit community structure and reveals Lime-resolved representational changes whose geomelric and entropy-based summaries align with linear probes, time-step ablations, and label state mutual information.
2
Related Work
Existing methods for interpreting RNNs can be categorized into performancc--rrentee and application- oriented post-training analyses. Performance-oriented approaches focus on evaluating network-level perfor- mance by comparing architectural components and training parameters. For example, Chung et al. (2014) compared gated RNNs (c.g., GRUs and LSTMs), while Greff et al. (2017) conducted a detailed analysis of LSTM components. However, these studies primarily emphasize performance outcomes and provide limited insights into the hidden state dynamics and learned representations within RNNs.
In contrast, application-orientel approaches focus on visualizing and interpreting hidden state activations after training, often in the context of specific tasks. In NLP, Karpathy et al. (2015) overlaid activation maps on texts, revealing interpretable unit behaviors such as tracking text structure. Li al. (2016) used saliency heat maps to identify critical words in learned representations, while Strobelt et al. and Ming al. developed interactive tools to correlate hidden state patterns with phrases (Strobelt et al. 2018 Ming let al. 2017). Similar techniques have been applied to domains such as speech recognition (Tang et al. 2017). earth sciences (Titos et al., 2022), and medical applications (Kwon et al.) 2019). While these studies provide intuitive, task-specific insights, they often lack generalizability and do not capture training dynamics over time.
Other works have explored general RNN behavior using techniques such as Proper Orthogonal Decomposition (POD) to analyze Seq2Seq internal states (Su & Shlizerman, 2020) and PCA to link recurrent activations to model generalization (Farrell et al. 2022). However, these approaches focus on post-training analysis and do not visualize how hidden states evolve across both time-steps and epochs during training. To our knowledge, no existing methods provide a unified view of RNN hidden dynamics over temporal and training dimensions.
3 Background
PHATE is a data visualization technique that can capture both the local and global struc- ture of data using diffusion processes (Moon et al. 2019). The PHATE algorithm optimizes the diffusion kernel (Coffman & Lafon 2006) for the visualization of high-dimensional data. Let æ, be a point in a high- dimensional dataset. PHATE begins by computing the Euclidean distance matrix E between all data points, where Egg- = [x. - 2l2. These distances are then transformed into affinities using adaptive a-decay kernel Kg,a m,))) 2 exp (-(2 E (..)
PHATE:
)" + exp exp
(-(_.te)) E.; ))). which adapts to the data density around each (.()))
point and captures local information. The parameters (z;) and (xx)) are the k-nearest- distance of 2; and Σj, and a controls the decay rate. The affinities are then row-normalized to obtain the diffusion operator P D-1 that represents the single-step transition probabilities between data points, where D is a diagonal matrix whose entries are row sums of KG PHATE calculates the information distance between points based on their transition probabilities: distij = / log P log PJll2, where p captures the transition probabilities of diffusion process on the data over steps and and j are rows in the ma- trix. These distances are embedded into low dimensions using Multidimensional Scaling (MDS) (Ramsay)
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1966) for visualization. Local and global distances within the data's manifold represented in PHATE by multistep diffusion probabilities. The diffusion probability of each point captures its local context, enabling pairwise comparisons between all points (both neighboring and distant points) that represent the entire global context. For further details, see (Moon et al. 2019). We will use PHATE to embed RNN training dynamics, however we alter the initial graph construction to emphasize certain structures the data we wish to visualize.
M-PHATE: Gigante et al. (2019) model the evolution of the hidden units in feedforward neural network and their community structure using& multislice graph. Each slice corresponds to the network at an epoch during training, and the collection of graphs represents the dynamical system resulting from the evolution of the network's hidden states. Let be an FNN with total of m hidden units, and let F(r) be the representation of the network after being trained for T E fl....,n) n) epochs on the training data X sampled from a larger dataset II. The algorithm first calculates shared feature space using the normalized activations of all hidden units i {l...., m] on the input data, a 3-dimensional tensor:
F (Y)) P E. (
T(T,i,))
V [ ( (Yi)
where F (Ya): : Rd - denotes the activation of the i-th hidden unit of F for the 1thh sample from input data Y. Here, Y is subset of samples from the d-dimensional training data X, with an equal number of samples from each input class, and < | This activation tensor T is then used to calculate intraslice affinities between pairs of hidden units within an epoch 7 during the training, as well as the interslice affinities between a hidden unit i and itself at different epochs:
[T(, 11
K (r) intrastice (i.j) exp
(G.1)
( i) T(,, 1)1
K(0 interslice (T, v) exp
2
where a is the a-decay parameter, (rr)) is the intraslice bandwidth for unit in epoch 5, and e is the fixed interslice bandwidth. These matrices are combined to form an nm nm multislice kernel matrix K, which is then symmetrized, row-normalized, and visualized using PHATE in 2D or 3D.
4 Multiway Multislice PHATE
M-PHATE was shown to be a powerful tool for visualizing FNNs. However, to effectively visualize the evolution of RNNs' hidden representations, we need to consider hidden state dynamics across time-steps within the sequence and training epochs concurrently (Su & Shlizerman 2020). In RNNs, the output from previous time-steps fed as an input to current time-steps. This is useful in the treatment of sequences and building memory of the previous inputs into the network. The network iteratively updates hidden state h. At each time-step t, the next hidden state h+41 computed using the input I, and the current hidden state he Importantly, the network uses the same weights W and biases b for each time-slep. Thus the output y at time-step t is y1 = f(W h2 + b), where f is some activation function.
Let R()) be the representation of an m-unit RNN after being trained for E {1.....n) epochs on the training data X II. We denote R(7) (Yk) RR + R the activation of the i-th hidden unit of R time-step w E (1,...,s} $} in epoch for the 4tth sample of Y, where Y consists of samples from the training data X. We construct the 4-way tensor T using the hidden unit activations as a shared feature space, which we use to calculate unit affinitics across all epochs and time-steps. The tensor T is an n x m X p tensor containing the activations each epoch 7 E fl...... nn and time-step E (l.....s) of each hidden unit R, (i E (1,...,m). m]] with respect each sample Yg C X. To eliminate the variability in T due to the bias term
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K (T(w) intrastop (i,j)
Epoch
B163
E29
F153
Time-stop
E180
K interstep
(T,, w), (1,
Figure 1: Example schematic of the multiway multislice graph used in MM-PHATE for RNNs. The intra- step kernels represent the similarities between the graph nodes at the same time-steps. The inter-step kernels represent the similarities between the nodes and themselves at different time-steps and epochs.
b, we 2-score the activation of each hidden unit at time-step and epoch
Law (YW) RE
T(T,,i,,)
(1)
[R
We construct kernel over T utilizing prior knowledge of the temporal aspect of T to capture its dynamics epochs and time-steps. This constructed kernel, denoted K, represents the weighted edges in the multislice graph of the hidden units (Fig. 1. In this representation, each unit has two types of connections: edges between the unit to itself across epochs and time-steps and, within a fixed epoch and Lime-step, edges between a unit and its community the other units which have the most similar representation. The edges are weighted by the similarity activation pattern. We define K as a nsm nsm kernel matrix between all m hidden units all s time-steps all n training epochs. The ((--1))) 1)sm + (w 1)m + row or column of K refers to the juh unit at time-step in epoch т. We henceforth refer to the row ((T,w,j). and the column K(; (T,, In order to capture the evolution of hidden units of R across time-steps and epochs, while preserving the unit's community structure, we construct multiway multislice kernel matrix reflecting two types of connections simultaneously. Given the o-decay parameter a, the intra-step bandwidth for unit i at time-step w and epoch 7: (Tw,))) and the fixed inter-step bandwidth c, we define:
Intra-step affinities between hidden units i and at timo-step in epoch
K (TW)) Intra-step (i.j) exp (- || T(T,,,, T(T,w,))
(r.
Inter-step affinities between a hidden unit i and itself at different time-steps and epochs:
T(T,,,, (-1 2
T(n, v,
(1) iiter-step
= exx
The bandwidth (r.ua) of the a-decay kernel is set to be the distance of unit at time-step from epoch to its 4th nearest neighbor across units at that time-step and epoch: (rawd) da (T w, w). w(),

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where dz(2, Z) denotes the lg distance from to its t nearest neighbor in Z. We used k 5 in all the results presented. The use of this adaptive bandwidth means the kernel is not symmetric and thus requires symmetrization. In the inter-step affinities K inter-step we use fxxed-bndwwidhh Gaussian kernel 1 - Ew=1. Ei=r dT(TTT, ((),,,,)) (.) ((,,,)) the average across all time-steps in all epochs and all units nem of the distance of unit time-stop to its ken nearest neighbor among the set consisting of the same unit at all steps.
The combined kernel matrix of these two matrices contains step in cach epoch, such that the intra-step affinities form a block diagonal matrix and the inter-step affinities form off-diagonal blocks composed of diagonal matrices (Fig. A.1
and column for each unit at each time-
(T(w)) K
intra-step ((..))
if (T,w) (1, ((T,W) (n.
(((T,,,,),
(0) K inter-stop
if i-
(2)
otherwise
We symmetrize this final kernel as K' = (K + KT), and row-normalize it to obtain P D-'K', where a diagonal matrix whose entries are row sums of K' and which P represents a random walk over all units at all time-steps in all epochs, where propagating from one state another is conditional on the transition probabilities between time-step w in epoch and time-step v in epoch r). PHATE applied P to visualize the tensor T in two or three dimensions. This resulting visualization thus simultancously captures information regarding the evolution of the units across both lime-steps and epochs.
5 Results
We evaluate MM-PHATE in two stages. First, we use controlled synthetic benchmarks to test whether the embedding preserves known dynamical structure and remains informative under smooth state-space pertur- bations. This stage provides ground-truth anchor for interpreting the geometry and entropy summaries used later. Second, analyze task-trained RNNs (Arca2Bump neural spiking Chowdhury al. (2022) and HAR kinematics Reyes-Ortiz et al. (2012) to assess whether MM-PHATE reveals Baariing-reevaane representational changes across time-steps and training epochs.
Throughout, we compare MM-PHATE against standard embeddings (PCA, I-SNE, Isomap, LLE, UMAP). We additionally report M-PHATE as a qualitative baseline in Appendix A.4 However, because M-PHATE operates on a single time-slice and does not model within-sequence dynamics, it embods different mathe- matical object (units epochs rather than units time epochs), making direct quantitative comparison inappropriate.
5.1 Controlled Synthetic Benchmarks: Bifurcation Families and Smooth State-Space Warps
Interpreting RNN training dynamics from visualization is challenging because the latent geometry is typi- cally unknown, and many embeddings are dominated by activation magnitude or coordinate parameterization rather than the underlying dynamics. To test whether an embedding preserves dynamically meaningful struc- ture, we use controlled synthetic benchmarks built from low-dimensional bifurcation systems with known qualitative geometry.
We use two-dimensional canonical systems (Hopf and Pitchfork) generate noisy hidden- undt readouts across epochs and time-steps (Appendix A.5 These benchmarks are designed to address two complementary questions:
whether the embedding distinguishes dynamical-famill differences (Hopf vs. Pitchfork), and
2. whether the resulting geometric organization remains interpretable under smooth state-space warps.
We first use the Hopf bifurcation as an analytically interpretable anchor benchmark validate geometric and entropy interpretations in a setting with known latent phases. We then use broader synthetic suite test family-level distinctions and warp perturbations.
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Epoch
MAA PHATE 2
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TINE
UMAP
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10
7 30
Dynamic Intent
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HMA 2
PCk
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00
55
1.
MIN I
UNAP
Figure 2: Visualization of supercritical Hopf bifurcation dynamics and their embeddings. The first column summarizes the synthetic construction used for the Hopf anchor benchmark: top a represen- tative latent Hopf trajectory for the swept-4 (dynamic) group, plotted in (z, y. μ)) and colored by training epoch, illustrating the transition from a contracting fixed point (μ < 0) to stable limit cycle (μ 0) I crosses 0; bottom-the corresponding latent phase portrait in (z, v), showing reprosentative trajectories from the static control group (u = 1 across epochs) alongside the same representative dynamic trajectory, colored by unit group (static vs. dynamic). The latent trajectories are then mapped through a scalar hidden- unit readout to simulate the model activations used for visualization. Remaining columns show embeddings produced by MM-PHATE, PCA, t-SNE, and UMAP, applied to the full hidden-state tensor and colored by epoch (top row) or unit identity (bottom row). MM-PHATE recovers the underlying bifurcation geometry as single coherent post-bifurcate orbit, whereas conventional methods primarily reflect changes in activation magnitude.
Hopf anchor benchmark: validating geometric and entropy interpretations. In the supercritical Hopf system, the control parameter μ1 governs transition from contracting fixed point (μx 0) to loss of stability μ 0, followed by convergence stable limit cycle for μ with radius r" (p) Vr. This gives controlled analogue of training-phest motifs that often appear in RNNs: contraction, transition/en peaaatiionexpasioon. and stabilization on a low-dimensional manifold. For this Hopf anchor benchmark, we use two groups generated from the same supereritical Hopf system but with different control-paramett schodules: a dynamic (swept-p) group with units, for which z is linearly swept from to across epochs, and a static (fixed- u)) control group with units, for which j = -1 at epochs. Thus, only the dynamic group crosses the bifurcation, while the static group remains in the contracting fixed-point regime. This provides an internal control for distinguishing bifureation-driven geometric reorganization from changes driven primarily by readout magnitude coordinate effects.
Figure 2 shows this Hopf cinamic-us.-statee anchor benchmark and compares embeddings produced by MM- PHATE and representative baselines. Each point corresponds to a hidden unit at a specific time-step and epoch. MM-PHATE recovers the qualitative dynamical progression: post-bifurcation epochs align onto a coherent periodic orbit, epochs remain globally orderod, and units follow parallel trajectories on shared manifold. Importantly, MM-PHATE deemphasizes amplitude growth as nuisance dimension and organizes points by geometry and temporal progression of the underlying dynamical system.
By contrast, standard embeddings in the full comparison (PCA, t-SNE, Isomap, LLE, UMAP; see Ap- pendix A.5.4p largely encode the increase in oscillation amplitude as grows, yielding families of concentric or inflated loops across epochs. In this regime, the dominant embedding axis reflects readout magnitude rather than the shared periodic topology, obscuring the transition and mixing static and dynamic units. When tanh nonlinearity is applied the readout, these variance-driven distortions become more pro- nounced due to saturation, whereas MM-PHATE, remains qualitatively stable (Fig. A.24
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MA.PNATE
UnB 231
MM PHATE
MM PMATH
MMA-PHATE
Figure 3: MM-PHATE on controlled synthetic bifurcation warp scenarios (first two embedding coordinates). Columns (left to right) show: (i) Hopf vs. Pitchfork with matched sweeps and no warp, (ii) Hopf vs. Pitchfork with smooth state-space warps applied to both systems, and (iii) a warp-only control in which both groups share the same Hopf latent trajectories and μ schodule and differ only by whether a smooth state-space warp applied. Top row: points colored by epoch. Bottom row: points colored by unit index (legend indicates unit identity and group membership).
This Hopf benchmark provides controlled support for two claims used later in the paper. First, in this sctting, MM-PHATE more faithfully reflects the latent the compared standard embeddings. Second, the embedding-spon entropy summaries introduced below are interpretable as geometric markers of representational regime changes: in the Hopf experiment, MM-PHATE intra-step and inter-step entropy recover the expected phase transitions more clearly than the baselines (Figs. A.22 A.23 These synthetic validations motivate the use the same summaries in task-trained
progression than
RNNs.
5.1.1
Dynamics-famity differences and smooth warp perturbations
We next lest whether MM-PHATE can separate cinamical-fammily differences from smooth coordinate per- turbations when both are present. To do so, we construct three controlled settings (Appendix A..52): (i) Hopf vs. Pitchfork with matched μ sweeps and no warp, (ii) Hopf vs. Pitchfork with smooth state-space warps applied to both systems, and (iii) warp-only control in which two groups share the same Hopf latent trajectories and y schedule and differ only by whether a smooth state-space warp is applied.
Figure summarizes the three scenarios (see also the per-scenario appendix visualization suites in Figs. A.6 A.17 In the unwarped Hopf-vs.-Pitchhors setting, MM-PHATE separates the two dynamical familics while preserving their distinct progression structure across epochs and time-steps (Figs. A.9 When both systems are smoothly warped, the family-level distinction remains visible, indicating that the embodding is not driven solely by the raw coordinate representation of the latent trajectories in the tested construction (Figs. A.10 A.13).
In the warp-only control, the two groups share the same Hopf latent trajectories and schedule and differ only by the application of a smooth state-space warp. MM-PHATE separates the clean and warped groups, showing that it is not invariant to smooth warps a strict sense. However, the bustbbifuuration limit-cycle structure remains qualitatively preserved in the MM-PHATE embedding (including poe/ppott-bifurctton progression). whereas the warp more strongly distorts the geometry recovered by PCA and t-SNE (Figs. A.14 A.17 This behavior consistent with robustness of the qualitative geometric progression to smooth state-
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space warps under the tested conditions, while making clear that MM-PHATE does not satisfy strict warp invariance.
Taken together, the synthetic benchmarks establish & coherent scope for the downstream analyses: MM- PHATE can recover meaningful geometric organization in controlled systems, can preserve qualitative pro- gression under smooth perturbations, and can distinguish family-level dynamics from coordinate-wal effects in the tested settings. We now turn to task-trained RNNs to test whether these geometric signals align with dearnin-rreleaan changes in real models.
5.2 Neural activity
PANAANN
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Figure 4: Area2Bump: Visualization of a 20-umit LSTM network trained for 200 epochs. Visualizations are generated using MM-PHATE, PCA, t-SNE, Isomap, Locally Linear Embedding (LLEE,, and UMAP (left to right). Points are colored by epoch (top row) or time-step (bottom row).
We next apply MM-PHATE to RNNs trained on the Arca2Bump dataset, which contains spiking activity recordings from macaques during a reaching task [Chowdhury et al. 2022). We trained a single-layer, 20- unit LSTM to classify reach direction and analyzed how its hidden representations evolve over training. In an example session, the model reached a validation accuracy of 74%. Let T denote the activation tensor across epochs, time-steps, and hidden units (with an activation vector for each unit obtained across trials; see Appendix A..3 for implementation details). Each point in the visualization corresponds to a specific unit time-step epoch triple (Fig. in practice, we subsample epochs and time-steps reduce memory load.
We compare MM-PHATE to five standard Smbeddings-PCA. t-SNE, Isomap, LLE, and IMAA-and to M-PHATE (Appendix A.4), using the same set of points. For the baselines, we reshape T into matrix points and whose columns are trial-indexed activations, and then apply
whose rows are teettttme-steeppeee each method to this flattened representation.
Qualitatively, MM-PHATE organizes the activity into a single manifold that simultaneously preserves pro- gression across epochs (top row) and continuity across time-steps (bottom row). In particular, MM-PHATE reveals clear reorganization that becomes pronounced at later time-steps and emerges (around epoch 40 to 90), roughly coinciding with the main rise and stabilization of validation performance (Fig. 4.. By contrast, PCA and Isomap primarily capture a global drift dominated by high-variance direc- tions and do not clearly soparate training phases. t-SNE emphasizes local neighborhoods that are strongly driven by time-step variability and often yields discontinuous structure across epochs. LLE and UMAP preserve some within-unit trajectories but still do not clearly expose the mid-training transition visible in MM-PHATE. This motivates the quantitative evaluations below.
mid training
5.2.1 Neighborhood preservation
To quantitatively compare how well each embedding preserves the structure of the activation tensor, we com- pute neighborhood preservation between the original activation space and the low-dimensional embedding.

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For each point catnn--ste--ppolle triple), we identify its k-nearest neighbors in the original space and in the embedding space, and measure the fraction of neighbors that are shared (Appendix A.6j. We report the average overlap separately for: (i) intra-step neighborhoods (within a fixed epoch and time-step, comparing units), and (ii) inter-step neighborhoods (within a fixed epoch and unit, comparing time-steps).
Table D summarizes results for the Area2Bump L.TMM. MM-PHATE achieves the highest intra-step preserva- tion at informative neighborhood sizes (e.g., k 5, 10, 15), indicating that it best captures within-time-step unit communities. For inter-step neighborhoods, MM-PHATE yields the highest preservation for all but the smallest neighborhood size (k 5), where LLE is marginally higher. We note that intra-step preservation necessarily saturates when k 2 m (here m = 20 units), explaining the ceiling at k = 20 and k = 40; those rows serve mainly as a sanity check.
An additional variant omitting Isomap (due to memory constraints at larger sample sizes) is reported in Appendix Table 3 the relative ranking of methods remains unchanged. Overall, these results support the qualitative impression from Fig. 4 MM-PHATE preserves both wilhin-time-step unit neighborhoods and across-time trajectories more faithfully than standard alternatives.
Table 1: Neighborhood preservation of visualization methods applied to an RNN trained on Area2Bump.
MM-PHATE
PCA
t-SNE
Isomap
LLE
UMAP
Intra-step
0.621
0.417
0.440
0.290
0.293
0.291
Inter-step
0.794
0.709
0.685
0.770
0.812
0.749
10
Intra-step
0.757
0.659
0.669
0.549
0.550
0.546
10
Inter-step
0.820
0.680
0.684
0.778
0.798
0.796
15
Intra-step
0.852
0.835
0.834
0.802
0.806
0.806
15
Inter-step
0.834
0.639
0.657
0.758
0.769
0.800
20
Intra-step
1.000
1.000
1.000
1.000
1.000
1.000
20
Inter-step
0.853
0.613
0.640
0.750
0.755
0.804
40
Intra-step
1.000
1.000
1.000
1.000
1.000
1.000
Inter-step 0.826
0.671
0.675
0.636
0.626
0.671
5.2.2 Intra-step entropy and time-resolved information analysis
To probe how representational geometry changes within epoch, measure how dispersed the hidden units are at each time-slep in the embedding. For a fixed epoch and time-step wi, we take the embedded points { (one per unit) and estimate the differential entropy of this point cloud using a KDE-based estimator. We refer to this quantity as intra-step entropy. Importantly, this is an embedling-sppcc measure of geometric dispersion (not a direct estimate the Shannon entropy the representation distribution in the information-theoreti sense); we use as a proxy for how strongly units diversify vs. collapse at a given time- step, and we validate its interpretability in a controlled setting. In the Hopf bifurcation experiment ($A.5) MM-PHATE intra-step entropy accurately recovers known contraction pwaectiooo/xxpasseee /wwppppppnnnnn/eeeeeeeeo /exppnsiio/staaalizaaiii /stailiiationii regimes of the latent dynamics, whereas variance-dominaled baselines do not (Fig. A.22
On the Area2Bump LSTM, MM-PHATE reveals a structured temporal progression (Fig. Entropy is relatively stable across time-steps until approximately epoch 30. Between epochs 30 and 50, entropy decreases for earlier time-sleps while increasing for later ones, coinciding with the initial rise in training and validation accuracy. This pattern is consistent with the network compressing weakly informative early-sequence activity while expanding later representations that more strongly align with the task. From epochs 50 to 90, entropy rises across most time-steps, with later time-steps peaking near epoch 90. Around this point, validation accuracy plateaus and the train-test loss gap widens, indicating the onset of overfitting. Thereafter, entropy for early time-steps continues to increase while entropy for later time-steps declines, consistent with increasing representational variability in the early sequence that does not translate into generalization.
Lincar probes, timc-stop ablations, and mutual information. The intra-stop entropy trends Fig. 5 suggest systematic divergence between early and late time-steps: early states compress around epochs 30-50 50, later states expand and peak near epoch 90, and after overfiuting onset the early sequence
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MM-PHATE
PCA
Isoaa
0.9
4770
-10
000
12-
3.4
12-
3.6-
-08
770-
-08
3.0-
0.-
-04
400
0.7
7744
0.-
4770
20
200
12-
3.4-
12-
3.6-
-55
100
777-
3.5-
0.8
3.0-
0.8-
Val Trann
774
100
200
100
150
200
100
200
100
100
200
Epoch
Figure 5: Intra-step entropy of hidden units in embedding space at each time-step and epoch, compared to validation accuracy (top) and losses (bottom), for MM-PHATE and baseline embeddings.
becomes increasingly high-entropy while late-step entropy declines. To Lest whether this early/ /late divergence reflects task-relevant structure (rather than embedding drift or variance scaling), we directly quantify label
information in the recurrent state he using three complementary diagnostics (Fig. G Appendix A.7
(.. A.9): (i)
time-resolved linear probes, (ii) time-step ablations of the trained classifier, and (iii) time-resolved mutual information (MT) between labels and h,
Linear probes measure how linearly decodable the label is from hn at each time-step throughout training. Probe accuracy changes little before epoch 30, consistent with the largely stable intra-step entropy over the same interval. Over epochs 30-50, probe performance begins to separate by time-step: later time-steps improve steadily and reach a higher, more stable test plateau, while early time-steps exhibit only a transient increase (peaking around epochs 50-60) before degrading the train-test gap widens. Notably, the phase boundaries in probe behavior align with the intra-step entropy phases: after approximately epoch 90, the onset of pronounced carly time-step overfitting coincides with sharp risc in carly-step entropy, whereas the late time-step Lest plateau occurs in the same interval in which late-step entropy begins to decline.
Time-step ablations test which parts of the sequence the trained decision rule depends on. Masking late inputs (carly-only) yiclds an carly rise and mid-training peak near the onsct of overfitting, followed by decline, consistent with reliance on brittle early-sequence shortcuts that do not generalize. In contrast, masking early inputs (late-only) improves gradually but remains below intact performance. After roughly epoch 80, early-only and late-only accuracies converge to similar moderate level while the intact model remains substantially higher, indicating that strong final performance depends on the full recurrent trajectory: evidence accumulated over the sequence is expressed most. reliably in later hidden states, but cannot. be recovered from either segment alone.
Tese-resolved MI corroborates this picture by measuring total label information present he independent of the readout. MT increases over training for both early and late segments, but remains consistently higher for later time-steps and reaches a larger final value. Crucially, the post-overfitting regime shows decoupling for early time-steps: intra-step rntropy continues to rise while probe accuracy and MI do not,, indicating that the growing early-step dispersion reflects label-irrelevant variability rather than useful task information.
Taken together, probes, ablations, and MI validate the interpretation of MM-PHATE intra-stop entropy as a meaningful geometric proxy: later time-steps progressively concentrate stable, task-relevant information, whereas early time-steps ultimately accumulate high-entropy variability that does not support generalization.
Comparison with baseline embeddings. Repeating the intra-step entropy analysis using PCA, I-SNE, Isomap, LLE, and UMAP (Fig. (5) yields substantially weaker alignment with probes, ablations, and MI. In particular, the basclines do not clearly reproduce the carly compression / late expansion regime or the later divergence between early vs. late time-steps; their entropy changes are delayed and/or /r dominated by global geometric effects. This mirrors the controlled Hopf bifurcation experiment, where standard embeddings fail
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Classification accuracy (test)
Mean mutual information
Mean probe accuracy
0.4
0.6
0.6
0.5
(train)
6.1
0.4
0.2
0.0
Late (train)
Late (Des))
0.3
-de-
Early only Date zeroed)
Late only (early zarced)
0.0
Early timesteps (0-340)
Late timesteps (350-390 5991
50
150
200
100
150
200
0
50
100
150
200
Epoch
Epoch
Epoch
Figure 6: Combined analysis of time-resolved representations. Left: Lincar probe accuracy for early and late time-steps on train and test sets. Middle: Time-step ablation accuracy for intact sequences, early- only sequences, and late-only sequences. Right: Time-resolved mutual information between labels and hidden states. All three diagnostics support the intra-step entropy interpretation: task-relevant information migrates toward later time-steps over training, while rising carly-step dispersion after overfitting reflects label-irrelevant variability rather than improved generalization.
to recover the known entropy transitions. In contrast, MM-PHATE provides a time-resolved view in which entropy trends, separability in the embedding, probe performance, ablation performance, and MI jointly track how different segments of the input sequence contribute to learning and overfitting.
5.2.3
Inter-step entropy
(a)
(b)
Cluster Center (Sillouutte Score: 0,77)
Inter-ttep Entropy
1.3-
-088
1.0
pocuraty
155
18
16
Hidden Ure
1.00
1.25
Cluster
Cluster
Cluster
006
Entropy
1.25-
7.1
-0.
0.5
Cluster
1.15-
1.20-
train
-02
50
100
150
1.10-
0.
1.05
Epoch
Epoch
100
(c)
(d)
Epoch
CM for Claster 6, Training
CM for Chuster 1, Training
CM for Cluster O Validation
for Cluster Validation
10
10
Adhaal
Predictd
Prediced
2 3
2
3
Figure 7: a) Inter-step entropy of each hidden unit across epochs, alongside model accuracy (left) and loss (right). b) Clusters of hidden units from inter-step entropy trajectories across epochs. Left: trajectories of units in cluster (12 units) and cluster (8 units). Right: cluster center trajectories across epochs. c d) Confusion matrices of sub-networks constructed from each cluster on training and validation data.
The analyses above characterize how representations evolve across unils at fixed time-step. We now ask complementary question: how temporally selective is each unit across the sequence? Whereas intra-step entropy summarizes diversity across units at given (r,w), inter-step entropy measures (for each unit) how strongly its representation differentiates among time-steps within an epoch, revealing temporal sensitivity.
Inter-step entropy reveals two robust groups of units (Fig. 7 One subset maintains higher inter-step entropy throughout training, peaking around epoch 80, indicating sustained temporal differentiation. The other subset shows pronounced decline in inter-step entropy later in training. indicating a collapse in temporal selectivity. To formalize this distinction, we apply time-series k-means with Dynamic Time Warping
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(DTW) distance [Bringmann et al. 2023) to the inter-step entropy trajectories (Fig. [p), so that units are grouped by their lemporal pallern of entropy changes even when the precise epochs of increase or decrease are slightly shifted. We emphasize that this DTWW-based clustering is used descriptively to summarize the heterogencity already visible Fig. 7. and our main conclusions do not depend on a specific choice of cluster number or assignments. Sub-networks constructed from these clusters exhibit. large performance differences (Fig. 7 d): the model using only the tigh-temporal-electiitty cluster achieve substantially higher training cluster, confirming that the
and validation accuracy than the model using only the low-temporal-selecee temporally differentiating units carry the most task-relevant signal.
This unit-level decompasition complements the intra-step entropy, probe, ablation, and MI results. To- gether they indicate that the example learning is accompanied by (i) increased task-relevant structure at later time-steps and (ii) an emerging specialization of hidden units, with a subset sustaining temporal se- lectivity while another subset becomes less informative later training. Among baseline dimensionality reduction techniques, PCA, t-SNE, Isomap, LLE, and UMAP fail to recover this functional separation or the associated training-phase transitions (Fig. A.43). In contrast, MM-PHATE preserves temporal continu- ity while exposing distinct unit-level roles, yielding coherent view of representational reorganization during learning.
Results for additional architectures (GRU, Vanilla RNN) and LSTM sizes (10-50 50 units) are provided in the appendix and show similar qualitative trends.
5.3 Analysis with Human Activity Recognition Model
To test whether the training-phann phate structure observed in Arca2Bump generalizes beyond neural spiking data, we trained a 30-unit LSTM for kinematics- based Human Activity Recognition (HAR) (Reyes- Ortiz et al. 2012). The model was trained for 1000 epochs and reached a final validation accuracy 84%. We applied MM-PHATE the full hidden-state activation tensor and repeated the intra-step and inter-step analyses from Section 5.2
(a)
(b)
setuate Eneagy
ACCURCC
1100
MM PHATE
-000
Timlo 200
-04
ISLOVATTE
MALOHATE
Brber shep Erergyy
MILINATE
MILPHATE
Epech
Epach
Figure 8: a) MM-PHATE visualization of a 30-unit LSTM network trained on HAR data, colored by epoch (Ieft) and time-step (right). b) Intra-step entropy, c) inter-stop entropy.
MM-PHATE resolves distinct training phases in within-sequence geometry. The HAR MM- PHATE embedding exhibits an ordered training trajectory (Fig. Sp. left): epochs progress smoothly along a coherent manifold rather than drifting arbitrarily. When colored by time-step (Fig- Bh, right), the within- sequence flow is not stationary across training, suggesting that the mapping from time-step progression to motion the representation manifold changes with learning. To quantify this effect, compute two epoch-wise metrics on the 3D embedding (Appendix A.14)): signed flow alignment centroid (direction of within-soquence progression) and epoch-to-epoce change magnitude (magnitude of temporal-goometrie reorganization between consecutive sampled epochs). Both metrics identify an early transition: the signed flow changes sign (negative to positive) around epoch ~ 7, coinciding with a spike in the change magnitude and sharp increase in training/ iianinn/valddatioon /aaiidaaion accuracy (Figs. A..77 A.49). This alignment supports that MM-PHATE captures a learning-relevann reorganization of within-sequence dynamics rather than a purely geometric artifact.
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Intra-step entropy shows plateau regime aligned with performance gains. We next mea- sure intra-step entropy to summarize how dispersed hidden units are within each time-step and epoch in embedding space (Fig. p in which we observed a transition around epoch 7 corresponding to the reorga- nization of within-sequence dynamics (Fig. [A500). Across training, later time-steps exhibit a charactoristic plateau followed by a drop in entropy, while earlier time-steps lend to increase. Importantly, the entropy drop aligns with an increase validation performance, suggesting that the late-step contraction reflects a useful compression of representations into more task-aligned, stable geometry. This interpretation is con- sistent with our controlled Hopf benchmark, where MM-PHATE intra-step entropy tracks known contrac- tion prappansioo/saabiiiaaiiiii regimes (Section A.5 providing calibrated link between entropy transitions and underlying dynamical organization.
Inter-step entropy indicates increasing temporal specialization during the same regime. Inter- step entropy (Fig. ( provides a complementary unit-level view by measuring how strongly each unit dif- ferentiates among time-steps within an epoch. During the interval in which late-step intra-step entropy drops and accuracy improves, many units show increased inter-step entropy, indicating stronger temporal selectivity. Together, the plateau drop pattern in intra-step entropy and the concurrent rise in inter-step entropy support a coherent picture: as learning progresses, the network both (i) consolidates later-step rep- resentations into a more compact, task-relevant geometry and (ii) increases the temporal specialization of subset of units, yielding clearer division of roles across the sequence.
Relation to expansion dynamics and comparison to Arca2Bump. The HAR results provide a time-resolved instantiation of the expansion expansion-compression compression view of representation learning in trained RNNs (Farrell et al. 2022. Here, MM-PHATE, makes the phases explicit: in within-sequence flow (captured by embedding-definee metrics) is followed by an entropy regime in which later time-steps increase, exhibit a plateau, and then compress, and this compression aligns with improved generalization and increased temporal specialization.
early geometric reorientation
Area2Bump does not show a late-step plateau-drop in intra-step entropy that co-occurs with improved validation performance; instead, late train- ing is dominated by rising early-step entropy that is decoupled from label information (probes consistent with bbbel-ireeleannt variability associated with overfitting rather than consolidation. A plausíble contrib- ulor is intrinsic data complexity relative to model capacity: HAR is substantially lower-dinenssonall than Area2Bump (6 vs. 35 PCs to explain 95% variance; Fig. A.2), making it easier for fixed-size LSTM to concentrate decision-relevant structure into a compact representation. Accordingly, the same MM-PHATE- based diagnostics highlight different regimes: HAR exhibits a performance-aligned late-step entropy drop together with increased inter-step entropy for many units, whereas Area2Bump shows weaker evidence of such consolidation and late rise in early-step dispersion that does not translate into generalization.
This behavior differs qualitatively from Area2Bump (Section 5.2
Overall, these results extend our findings from controlled dynamics (Hopf) and neural spiking data (Area2Bump) to a distinct kinematic domain, showing that MM-PHATE can resolve multiple learning- relevant phases of within-sequence organization and that these phases can be quantified directly from the embedding.
6 Conclusion
We introducod MM-PHATE, a visualization framework for RNNs that embods hidden-state dynamics jointly across lime-steps and training epochs via a multiway multislice graph. This construction preserves within- time-step unit communities and across-time trajectories in single coherent manifold in the settings studied here, addressing common failure modes of standard embeddings that collapse or distort one or more of these axes.
Across controlled synthetic benchmarks and real RNN applications, MM-PHATE produced geometry that aligned with known or independently measured structure. In synthetic bifurcation systems, MM-PHATE recovered the Hopf contraction contractioo//expaniion. aaaansionnstabbizatiin progression more faithfully than the compared stan- dard embeddings and remained informative under smooth state-space warp perturbations, while preserving
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visible distinctions between dynamical families (Hopf vs. Pitchfork) when both dynamics-type and warp differences were present. These controlled results provide empirical support for the geometrie interpretations used later in the paper, but do not constitute a formal invariance guarantee.
On task-trained RNNs, MM-PHATE achieved the strongest neighborhood preservation on Area2Bump and revealed time-resolved maiiingg-paaee changes that aligned with probes, ablations, and mutual information, while inter-step entropy identified functionally distinct unit groups with different temporal selectivity. On HAR, MM-PHATE generalized these observations and exposed pronounced training-phase transition (including a reversal of within-sequence flow direction) with coordinated intra//inter-ster entropy changes that coincided with performance improvements, suggesting task-dependent consolidation consistent with dataset complexity differences.
Limitations and future directions. MM-PHATE encodes contimuity across time and epochs; highly non-smooth training dynamics (c.g., abrupt restarts or extreme learning rates) may reduce interpretability, While our synthetic benchmarks cover multiple bifurcation families and smooth state-space warps, extend- ing these evaluations and developing theoretical guarantees (e-g., under classes of smooth equivalences perturbations) remains an important direction. The nsm nam kernel can also limit scale, but PHATE is robust to subsampling and scalable approximations (e.g., subsampling, coarsening, partitioning) are prac- tical routes forward. Extending the multiway multislice construction to richer architectures (e.g., deeper recurrent stacks or attention- based models) an important direction for future work (Katharopoulos et al.).
Acknowledgments
N.M. was funded as fellow by the Kavli NeurData Discovery Institute. G.M. was supported by NSF grant numbers CCF 2217058 and 2403452. A.S.C. was supported in part by NSF CAREER award number 2340338.
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A Appendix
A.1 Datasets
We employed two datasets (Arca2Bump as neuroscience-motivaeed benchmark of population dynamics; HAR as a real-world sequential classification dataset).
1. Arca2Bump is a part of the well-known "Neural Latents Benchmark" Challenge Pei et al. 2021 in the neuroscience community. The Area2Bunp dataset (Chowdhury et. al.] 2022) consists of neural activity recorded from Brodmann's area 2 of the somatosensory cortex while macaque monkeys performed a slightly modificd version of standard center-out reaching task. Dataset license: CCO 1.0. According to the authors, data was collected consistently "with the guide for the care and of laboratory animals and approved by the institutional animal care and committee of
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Northwestern University under protocol #1500000367'. This dataset, collected from monkeys thus does not contain personally identifiable information. Since it is neural data, we do not consider it. to be offensive content.
Data Statistics: The dataset includes 193 samples, split into 115 training samples and 78 testing samples. Each sample consists of 600 time-steps with 65 features per time-step. For the training set, the mean values across features range from 0.0001 to 0.03, with standard deviations between 0.001 and 0.02, and maximum values ranging from 0.003 to 0.12. In the test set, the mean values range from 0.00008 to 0.03, standard deviations from 0.0007 to 0.018, and maximum values from 0.0007 to 0.13.
2. The HAR dataset was originally introduced in Anguita ct al. (0013) which has been cited 3k times, and is also part of the UCI ML repository (where it has been viewed 196k times). The Human Activity Recognition (HAR) Using Smartphones dataset (Reyes- Ortiz et al. 2012), kinematic record- ings of 30 subjects performing daily living activities with a smartphone embedded with an inertial mcasurement unit. Dataset license: CC BY 4.0. Information relating to participant consent was not found relating to this dataset. This dalaset does not reveal participant name or identifiable information. The kinematics contained in the dataset are not considered offensive content.
Data Details: The dataset includes six activity classes (e.g., Walking, Sitting) based on accelerometer and gyroscope data sampled at 50 Hz. The sensor data has been pre-processed with noise filtering and separated into gravitational and body motion components. Data is windowed into 2.56-second segments (128 data points) with 50% overlap, resulting in 561-dimensional feature vectors per win- dow. The datasct is split into 70% training (21 subjects, 7352 samples) and 30% testing (9 subjects, 2947 samples).
Data Statisties: For the training set, mean values across features range from -0.0008 to 0.8, with standard deviations between 0.1 and 0.41, and values spanning from -5.97 to 5.75. For the test set, mean values range from -0.013 to 0.8, with standard deviations between 0.095 and 0.41, and values spanning from -3.43 to 3.47.
A.2 Model Training
We used TensorFlow's Keras APT for all model training and validation.
The network in Section 5.2 was trained as follows. The Area2Bump dataset was randomly split into training and validation subsets containing an equal number of samples for each input class with an 8 to 2 ratio. Additional samples that would have made the training data uneven were added back to the validation subset to utilize all available samples. The network consists of an LSTM layer with 20 units. This is followed by a Flatten layer that converts the LSTM's output into a one-dimensional vector. Finally, Dense layer with 8 units and a softmax activation function produces the output for the 8-class classification tasks. The network was trained with batch size of 64. We used an Adam optimizer with a learning rate of 1 During the training process, we recorded the activations from the LSTM layer into the activation tensor T for visualization.
The network in Section 5.3 was trained as follows. The HAR dataset was preprocessed and split by the authors into training and validation subsets according to the subjects with to 3 ratio. The network consists of an LSTM layer with 30 units and Dense layer with 6 units and softmax activation function produces the output for the 6-class classification tasks. The network was trained with batch size of 32. We used an Adam optimizer with a learning rate of 2e During the training process, we recorded the activations from the LSTM layer into the activation tensor T for visualization.
A.3 Implementation of visualization methods
Common tensor and reshaping. For each trained network or dynamical system, we record hidden activations into a tensor with axes (epoch 7) x (unit i) x (time-step w) x (trial/sample index). For baseline embeddings that operate on a matrix, we reshape the sampled tensor into
T Thaa E IR. Np
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E1151
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Epoch and Time ste--tep Time-step
E2S1
E282
E283
E3S1
E382
E3S3
K((T, w, i), (n, ()
K
intrastep
K(I) Kinterstep ((T, w), (n, v))
Figure A.1: Example schematic of the multiway multislice kernel used MM-PHATE for RNNs. The intra-slep kernels represent the similarities between the graph nodes at the same time-steps. The inter-step kernels represent the similarities between the nodes and themselves at different time-steps and epochs.
where each row corresponds to one mit--timn-step-ppochh triple and the p columns index trial-wise activations, as in the main text.
Hopf bifurcation dataset. For the Hopf experiment, the main activation tensor is
trace_datt data E ((ET)xXX
where is the number of epochs, T the number intrinsic time-steps per epoch, H the number of hidden units, and S the number of samples (trials) per unit. Unless otherwise noted, Hopf embeddings use all epochs and time-steps; we do not subsample т or w for MM-PHATE, PCA, t-SNE, or UMAP. LLE and Isomap operate on random subset of points, but still draw from the full (T,w, i) grid.
Area2Bump subsampling used for MM-PHATE and most baselines. Due to memory constraints, for Area2Bump we compute MM-PHATE, as the first 29 epochs followed by every 5th epoch thereafter (up to 200 total epochs), and (ii) time-steps sampled uniformly over the 600-step sequence using 100 evenly spaced indices. Unless stated otherwisc below, I-SNE, LLE, UMAP are computed on this same sampled tensor TMM-
subsampled tensor TMM defined by: (i) epochs sampled
HAR subsampling used for MM-PHATE. For HAR, we sample epochs as the first 29 epochs followed by every 10th epoch thereafter (up to 1000 epochs) and compute MM-PHATE on the resulting tensor. For bascline embeddings we reshape the corresponding sampled tensor into Tiat in the same way.
A.3.1 MM-PHATE
For Area2Bump and HAR, MM-PHATE is applied to the sampled tensors described above (Area2Bump: Taи; HAR: the sampled HAR tensor). We construct the multiway multislice kernel over units, time-steps, and epochs and then run PHATE via the M-PHATE package with n_ n_components 3.
For the Hopf bifurcation dataset, we apply MM-PHATE directly to trace_datt data of shape (ET, (no epoch time-step subsampling), again with n components 3.
A.3.2 PCA
Due to PCA's low computational demand, we did not. subsample any data before applied PCA using sklearn. www.mpositiii..........
Figure A.2 reports how many principal components are required to explain 95% of the variance for A Area2Bump and HAR.
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Scree Plot Area2Bump
Curnulative Explained Veriance
Scree Plitt HAR
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Figure A.2: PCA on the two datasets, Area2Bump requires 35 PCs (left) to explain 95% of the variance, whereas HAR requires 6 PCs (right).
For the Hopf dataset, we flatten prace_data to
X
by stacking all (r,w, i) combinations and treating the S samples as features. We mean-center X across samples and fit PCA with n_ components 3 to obtain a 3D embedding used in the Hopf figures (epoch-, time-, unit-, and group-colored views).
A.3.3 t-SNE
For Area2Bump, we first reduce Tmat to 15 principal components (explaining 99.93% of the variance) and then run Barnes-Hut t-SNE with n_components 3, perplexity 50, and __iter 2000 using sklearn. www.fold. TSNE.
For Hopf, we the same t-SNE configuration the centered matrix X R(ETH)xS described above (no epoch timc-step subsampling). Because ETH is on the order of 8 x 104 in our configuration, this run is heavy but tractable on our cluster.
A.3.4 Isomap
Due to scalability constraints, Isomap is only applied subsampled data.
For Arca2Bump, we build a more aggressively subsampled tensor Tiso by sampling (i) epochs as the first 29 epochs followed by every 10th epoch thereafter, and (ii) 50 evenly spaced time-steps from the 600-step sequence. We then reduce Tiso to 15 PCs and apply sklearn. manifold. Isomap with n neggghhoors = 30 and n_ n_components = 3.
For Hopf, we do subsample epochs or time-steps, but we do randomly subsample points across the flattened matrix X (ETH))S to form
Xsub N
where Nou min(10,000, FTH) (with fixed random seed). We then run Isomap (n meihhbos 30, n_ components 3) on Xsub- The label vectors for epoch, time-step, unit, and group are restricted to the same subsample to produce unil-, epoch- and group-colored Hopf panels.
A.3.5 Locally Linear Embedding (LLE)
For Area2Bump, LLE is computed the sampled tensor TMM used for MM-PHATE. We re- shape to Tmat and apply LocallyLneerFmbdddng (n neighbors = 100, components = 3, method = "modified").
For Hopf, as with Isomap, we use the random subsample Xsub size Nsub drawn from the full flattened matrix X. We then run LocallyLinerrEneeddnn (n neighbors 20, n_components 3, method "modified") obtain a 3D embedding on the subsampled points, and use the corresponding epoch ppoch/time//nit/groupp labels to generate the Hopf LLE panels.
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A.3.6 Uniform Manifold Approximation and Projection (UMAP)
For Area2Bump, UMAP is applied to Tmat formed from TMM- We use umap.UMAP min_ dist 0.1, , n_ components 3, Euclidean metric, and random_stane
UMAP with n_neighbors 50, 24.
For Hopf, UMAP run on the full centered matrix X R(ETH)xS with n o__neigborrs 30, min_ dist 0.1, c_comoonents 3, and Euclidean metric. No epoch time-step subsampling is used here; each (T,w,i) point is included.
A.4 M-PHATE as Qualitative Baseline on single time-step
To provide qualitative point of reference, we also apply M-PHATE to the Hopf bifurcation experiment and the Arca2Bump LSTM. Importantly, M-PHATE operates on single snapshot of network activity: it embeds units based solely on their activations at one time-step per epoch (lypically the final step). We select the final intrinsic time-step w = T for all epochs, yielding a tensor
trace_lass ExH8
and flatten it to
Xast E (E(E))SS
After mean-centering across samples, run PHATE components 3, knn = 30) to obtain a 3D embedding of epoch-unit pairs at the final time-step. This differs fundamentally from MM-PHATE, which embeds the full multislice tensor (units x time x epoch) using diffusion process that jointly couples temporal and epoch-wise transitions.
Why M-PHATE is not an appropriate comparison. Because M-PHATE ignores all but a single snapshot of the time-steps, it discards the within-epoch temporal dynamics that are essential for under- standing recurrent computation. RNNs process input sequences through evolving trajectories, and many of their representational transitions-incluee within epoch's trajectory rather than only at its final state. Consequently:
contraction, expansion, or bifurcation-like behavior- occur
M-PHATE cannot represent how individual hidden units transition across time-steps.
It cannot reveal dynamical geometry such as pre-bifurcation collapse, transition structure near μ 0, or post-bifurcction limit-cycle organization.
embeds a different mathematical object (units x epochs), whereas MM-PHATE embeds units X time epochs; the two inhabit incompatible geometric spaces, making direct quantita- tive comparison inappropriate.
Thus M-PHATE serves only as a qualitative bascline showing how standard diffusion-basod embeddings behave when restricted to a single time-slice, rather than an algorithm targeting dynamical systems.
Hopf control experiment. Figure A.3 shows M-PHATE applied to the last time-step of the Hopf dataset across epochs and units. While it captures a smooth trajectory as y increases, it misses the characteristic change in withhin-ppch flow geometry that differentiates pre- and post-bifurcation dynamics (e.g., noise- dominated collapse growing limit cycles). This demonstrates that restricting to the final state obscures the dynamical phenomena MM-PHATE is designed to uncover (Fig. A.18
Arca2Bump LSTM. Figure A.4 shows the same M-PHATE procedure applicd to the 20-unit LSTM trained on the Area2Bump dataset. The structure is smooth across epochs, and hidden units display modest differentiation; however, the embedding lacks any representation of the sequential processing dynamics that occur within each trial. As result, M-PHATE provides a useful snapshot-level visualization, but cannot address questions related to temporal geometry, representational flow, or information propagation that MM- PHATE is explicitly designed to capture.
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Figure A.4: M-PHATE applied to the Area2Bump LSTM (last lime-step across epochs). Points represent. hidden units at the final time-step each epoch, colored by epoch (left) and by unit identity (right)). The visualization captures smooth epoch progression but omits the temporal dynamics that are central to recurrent computation.
A.5 Controlled Synthetic Bifurcation Benchmarks with Smooth Warps
Scope and motivation. This appendix documents the controlled synthetic benchmark suite used to ad- dress reviewer concerns about (i) multiple bifurcation families, (ii) smooth state-space perturbations (warps), and (iii) settings in which faramics-family differences and smooth warps are both present simultancously. These experiments are intended as empirical stress lesls of MM-PHATE's geometric behavior in controlled settings, not as formal guarantee of invariance or equivalence under smooth transformations. In particular, our state-warp experiments are motivated by recent work on dynamical comparisons under smooth trans- formations and alignment alignmen/eequivalenee /equivalence viewpoints (Sagodi Park 2025 Chen al. 2024 Friedman al. 2025), but evaluation remains qualitative/empiriial and limited to the tested benchmark conditions.
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Each syuthetic dataset contains two latent groups (A and B), each governed by a twooddimensional ODE with the control parameter for Published in Transactions on Machine Learning Research (02/2026)
A.5.1 Data Generation
We use controlled low-dimensional synthetic dynamncal systems to test whether MM-PHATE recovers geo- metric organization associated with bifurcation structure, and whether these patterns remain interpretable under smooth perturbations of the latent coordinates. The benchmark suite is generated from canonical two-dimensional bifurcation systems (Hopf and Pitchfork), with optional smooth state-space warps before the ladent-to-hidden readout.
The benchmark design probes three complementary questions: (i) whether embeddings distinguish qualita- tively different bifurcation familics (Hopf vs. Pitchfork), (ii) whether this distinction remains visible under smooth stale-space warps, and (iii) whether the method preserves the qualitative bifurcation progression in a warp-only control (Hopf clean vs. Hopf warped) where the two groups are identical before warping and read- out. In the warp-only control, we simulate a single latent trajectory once, copy it to both groups, and apply the warp only to one group, the pre-warp latent trajectories are exactly identical by construction. This removes latent-initialization and trajectory-generatioon confounds when assessing sensitivity to coordinate changes.
control parameter μ varying across training epochs. For epoch {O....,, 1}, group g E {A, B} follows a linear schodule
Ng Mg.start
(Mggend (g,start);
(3)
so that different scenarios can implement matched sweeps (e.g., Hopf vs. Pitchfork with μ -1 - 2), within-family parameter shifts (e.g. -1 -1 - 3), identical schedules for warp-only controls.
Latent dynamics. We use two canonical supercritical bifurcation systems. For the Hopf benchmark, we use the supercritical Hopf normal form
à pat wa (x² + y² )r,
(4)
ý war + jy - (2² + y y.
(5)
where w the angular frequency (always here for simplicity). For μ < 0, trajectorics contract to the fixed point at the origin; for a 0, trajectories converge to stable limit cycle.
For the Pitchfork benchmark, we use supercritical pitchfork in with a linearly stable auxiliary dimension y:
á = μm 33 j = - Ayy with À >> O. This produces qualitatively different post- bifurratiin organization from Hopf (branching in æ rather than rotational limit-cycle dynamics).
(6)
(7)
Epochs and time-steps. Within each epoch t, the latent system is integrated for 8 time-steps at fixed (T) using a fourth-order Runge Kutta method with step size At. Thus each epoch contributes a short trajectory segment under constant control parameter, and the full dataset traces a progression across epochs as μ sweeps through the bifurcation regime.
Optional smooth warps. To test robustness to smooth perturbations of the latent coordinates, we apply an optional smooth state-space warp (a near-identity diffeomorphic transform RR) to each epoch trajectory before readout. In the implementation used here, the state warp is
( v) (sr kay, y + ka²).
(8)
with small x (default 0.15), yielding smooth coordinate deformation that preserves local continuity while changing the raw coordinate representation.
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Readout mapping (latent hidden-unit activation). At each time-step in epoch 5, the latent state (x, v) is mapped to scalar activation by combining shared linear projection and a radial component:
h a aww [zr,0 Vr + TTrw
Tr,
x + V
(9)
where w E IR² is a shared random projection direction (sampled once per dataset and shared across both groups), and o, control the relative contributions of signed projection and radius. Optionally, the scalar readout is passed through tanh(-) to mimic an RNN nonlinearity.
Hidden units and group structure. Each group contributes a fixed number of hidden units (typically for group A and for group in the minimal benchmark suite). Within group, units are generated by replicating the same group-level scalar readout and adding small i.id. noise, with optional per-unit gain and bias jitter. This produces light but non-identical unit trajectories while preserving the underlying group-level latent dynamics.
Samples. We generate p independent samples (trials), each with its own initial condition and process noise realization. All samples within scenario share the same ODE specification, parameter schedule(s). and warp scttings, but produce distinct trajectorics. In the warp-only control, the two groups share the same simulated latent trajectories before warping by construction, and differ only through the applied warp and downstream readout replication.
Tensor structure and default configuration. The final activation tensor matches the notation of Table
T E (mx))xxxp
where the first dimension indexes flattened epoch- timeehttm--tepp pairs (T, w), m is the total number of hidden units across both groups, and p is the number of samples. In the benchmark configuration used throughout the synthetic suite unless otherwise stated, we use
101,
8 8 80,
= 6 (-33++
10,
with At 0.1 and linear sweep in μ across epochs (e.g., μt: :4-1-1 2). Figure A.5 illustrates the latent and readout construction for Hopf-based example.
The Hopf anchor benchmark (static vs. dynamic) used for entropy validation (Section A.5.4 uses different unit split (6 dynamic + 4 static) and is described separately there.
A.5.2 Scenario suite used in the main text
Table ssummarizes the three core synthetic scenarios used in the main-text comparison (Section 5.1.1 These scenarios isolate (a) dynamical-family differences, (b) smooth state-warp perturbations, and (c) the combination of both factors. Unless otherwise stated, all scenarios use matched epoch tiimestep settings, the same readout construction, and matched μ schedules across groups.
A.5.3 Synthetic-suite results: bifurcation families and smooth state-space warps
Figure 3 (main text) summarizes the three core synthetic scenarios listed in Table 2 In this appendix subsection, we provide the corresponding per-scenario visualization suites used to support that summary. For readability, we include the full 2-D/3-D embedding views colored by epoch, timestep, group identity, and unit index, together with latent phase portraits and readout traces.
Across these settings, MM-PHATE exhibits three qualitative behaviors relevant to the reviewer concern: (i) sensitivity to differences in latent dynamical family, (ii) retention of a family-level distinction under smooth state-space warps, and (iii) preservation of progression structure even when the same latent dynamics are observed through different coordinates. In every composite embedding panel below, the three columns are MM-PHATE, PCA, and i-SNE (left-to-right).
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Figure A.5: Latent and readout dynamics of the Hopf bifurcation experiment. (Top left) Latent (x, y) trajectories of the Hopf system as μ moves from negative (stable focus) to positive (stable limit-cycle). illustrating the bifurcation. (Bottom left) Example hiddlen-umii-ggeeratnng trajectories in the 2D latent plane, illustrating how latent dynamics across epochs converted into group-wise readout trajectories. (Top right) The corresponding one-dimensional scalar readout obtained from h = a(w [z, vl) + before optional nonlinearity. (Bottom right) The scalar readout after applying tanh nonlinearity, which compresses largo-amplitude oscillations and highlights saturation in the cost-bifurcation regime.
Table 2: Controlled synthetic scenarios used to test dynamics-family differences and smooth state-space warps. Main-ton text results focus on state-warp settings (no time warp). Scenario
Group A
Group B Pitchfork
State warp none / none
What it tests Family-level dynamical
Hopf vs. Pitchfork Hopf (clean)
differences without dinate perturbation Whether family-level dif- ferences remain visible when both systems are smoothly warped Sensitivity to smooth coordinate perturbation when latent dynamics are otherwise identical
Hopf vs. Pitchfork Hopf (warped)
Pitchfork
yes yes
Hopf clean vs. Hopf warped (warp-only control)
Hopf
Hopf (same latent no yes trajectories before warp)
(1) Dynamics-family differences without warps. In the unwarped Hopf-vs.-Pitchfork setting, Group A is the Hopf bifurcation system and Group B is the Pitchfork bifurcation system (consistent with the scenario name hopf seaff_if___p__ bif__pitchforr__bbt.. httpe_hforobbb___oooooo______e _______waa_ _o__wrp). MM-PHATE separates the two groups while preserving distinct progression structure across epochs and time-steps (Figs. A.6 A.9. The latent phase portrait confirms that the two groups differ the level of generating dynamics (rotational organization for Hopf versus branching organization for Pitchfork in this construction), while the readout. traces help verify that the observed separation is not explained by trivial monotonic scaling alone.
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Figure A.6: Hopf vs. Pitchfork without warps: 3-D embedding views (epoch, timestep, and group identity). From top to bottom: colorings by training epoch, within-epoch timestep, and group identity (A Hopf bifurcation, B = Pitchfork bifurcation). Fach image is three-column composite (MM- PHATE, PCA, t-SNE). These views assess progression organization and family-level separation in 3-D.
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Figure A.7: Hopf vs. Pitchfork without warps: 3-D unit-level organization and latent phase portraits. Top: embeddings colored by unit index. Bottom: latent phase portraits for Group A (Hopf bifurcation; left panel) and Group B (Pitchfork bifurcation; right pancl), i.c,, the ground-truth latent dy- namics used to generate the observations.
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1-NE 1
MIAPHATT 2
PCA2
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Group pitchlokk
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PCA1
1-SNE
Figure A.8: Hopf vs. Pitchfork without warps: 2-D embedding views (epoch, timestep, and group identity). These 2-D projections the compact counterparts of Fig. A.6 and verify that the same qualitative progression and family-level separation patterns remain visible after projection to two dimensions.
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MM.PHATE 2
Unit
PCA2
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Figure A.9: Hopf vs. Pitchfork without warps: 2-D unit-level organization and average readout traces. Top: embeddings colored by unit index. Bottom: average readout traces for groups A and B across global training time. Together with the latent phase portraits in Fig. [A.7] these panels support the interpretation that MM-PHATE is sensitive to qualitative generic amplitude growth.
differences rather than only
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(2) Dynamics-family differences with smooth state-space warps applied to both groups. When both systems are subjected to smooth state-space warps prior to readout, Group A remains the warped Hopf bifurcation system and Group B remains the warped Pitchfork bifurcation system (consis- tent with aopf_bif__pteeeeeeeeeeeppppppeee bif__pitcchfork_bitt while retaining epoch/Limg Limesstep progression structure (Figs. A.10 A.13). This indicales that, the tested conditions, the embedding is not simply memorizing raw latent coordinates and remains informative under smooth coordinate perturbations. The latent phase portraits and readout traces again provide the mecha- nistic context for interpreting the embedding geometry.
MM-PHATE still preserves a visible family-level distinction
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100
MM-PHATE
PCA2
L-SNE
MIA-PHATE
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40
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PCA2
I-SNE
MM-PHATE
PCA
t-SNE
Group (n=3)
(n=3)
MM-PHATE2
PCA2
1-NE
MM-PHATE
PCA1
Figure A.10: Hopf Pitchfork with smooth state-space warps: 3-D embedding views (epoch, timestep, and group identity). From top bottom: colorings training epoch, within-epoch timestep, and group identity (A warped Hopf bifurcation, B warped Pitchfork bifurcation). Each image is three-column composite (MM-PHATE, PCA, t-SNE). Despite coordinate warps, MM-PHATE continues to display organized progression and a visible family-level distinction.
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Unit
Unit (A)
Unit (A)
Unit
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MM-PHATE
PCA2
1-SNE
MM-PHATE
100
7000
7000
Epoch=
Epoch<T
4000
4000
3000
2000
2000
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1.5
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-15
Figure A.11: Hopf vs. Pitchfork with smooth state-space warps: 3-D unit-level organization and latent phase portraits. Top: embeddings colored by unit index. Bottom: latent phase portraits for Group A (warped Hopf bifurcation; left panel) and Group B (warped Pitchfork bifurcation; right panel). These panels help verify that the observed family-level separation is not driven by only small subset of units and remains interpretable relative to the underlying latent trajectories.
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100
HMM-HATE
PCA2
1-SNE 2
MM-PHATE
PCA1
t-SNE
MM 2 2
PCA2
USNE 2
MM-PHATE
PCA1
1-SNE 1
MIAPHATE 2
PCA2
1-SNE 2
Group pitchlokk
MM-PHATE
PCA1
L-SNE
Figure A.12: Hopf vs. Pitchfork with smooth state-space warps: 2-D embedding views (epoch, timestep, and group identity). The 2-D projections confirm that the family-level distinction and pro- gression organization remain visible in compact representation, rather than only in favorable 3-D view.
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2MM-PHATE2
Unit
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1-SNE
Unit (B)
Unit (8)
MM-PHATE
PCA1
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Epoch
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A Ahopt, ri3)
toithrork n=3)
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1.00
0.75
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Global time (epoch T
Figure A.13: Hopf vs. Pitchfork with smooth state-space warps: 2-D unit-level organization and average readout traces. Top: embeddings colored by unit index. Bottom: average readout traces for groups and B across global training time. These panels complement the 3-D views and support the claim that MM-PHATE remains informative under smooth coordinate warps in the tested synthetic suitc.
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(3) Warp-only control (same latent dynamics, different coordinates). In the Hopf clean-vs.-Hopf warped control, Group A is the clean Hopf system and Group B is the warped Hopf system (consistent with hopf big_ clean ve warped); both groups share the same latent trajectories and schedule before warping by construction. MM-PHATE still separates the clean and warped groups, showing that it is not invariant smooth warps a strict sense; however, preserves the qualitative bifurcation progression (including pre/ppost-bfurration organization), and in particular the post-bifurcation limit-cycle structure remains visibly preserved in MM-PHATE across the compared views (Figs. A.14 A.17 By contrast, PCA and t-SNE show stronger warp-induced geometric distortion, which makes the shared Hopf post- post-biffurcatin organization less clear. This is the key qualitative observation motivating the downstream entropy analyses: under smooth coordinate warping, MM-PHATE preserves qualitative dynamical geometry (notably the post- bifurcation limit-cyele organization and progression structure) despite lacking strict invariance, whereas PCA and t-SNE more strongly distort these features.
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PCA2
L-SNE
MMA-HATE
PCA
1-SNE
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50
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1-SNE
MM-PHATE
PCA
1-SNE
Group (n=3)
2
PCA2
L-NE
MM-PHATE
PCA1
1-SNE
Figure A.14: Hopf clean vs. Hopf warped (warp-only control): 3-D embedding views (epoch, timestep, and group identity). From top bottom: colorings by training epoch, within-epoch timestep, and group identity (A clean Hopf, latent dynamics before warping, the group separation here reflects sensitivity to coordinate warp rather than change in dynamical family. However, despite separating the clean and warped groups, MM-PHATE still preserves the qualitative post-bifurcation limit-cycle organization, whereas PCA and t-SNE show stronger warp-induced geometric distortion that obscures this shared Hopf structure.
warped Hopf). Because both groups share the same underlying
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Unt (A)
Unit (A)
Unit (8)
Unit
2
PCA2
L-SEE 2
MM-PHATE
PCA
1-SNE
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Rg
-15
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Figure A.15: Hopf clean vs. Hopf warped (warp-only control): 3-D unit-level organization and latent phase portraits. Top: embeddings colored by unit index. Bottom: latent phase portraits for Group A (clean Hopf; left panel) and Group B (warped Hopf; right pancl). The latent panel verifies that the underlying bifurcation progression is matched by construction prior to warping, making this direct test of warp sensitivity versus progression preservation.
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100
MMM-HATE
PCA2
1-SNE
MM-PHATE
PCA1
E-NE
MM 2HATHATE 2
PCA2
JSNEE
MM-PHATE
PCA1
t-NNE 1
MM.PHATE2
PCA2
1-SNE
Group hopt (nn3)
MM-PHATE
PCA1
1-SNE
Figure A.16: Hopf clean vs. Hopf warped (warp-only control): 2-D embedding views (epoch, timestep, and group identity). The same qualitative interpretation persists in 2-D: MM-PHATE dis- tinguishes the nordinate-warpel observation spaces while preserving the temporal temporal/progrossion progresssion organization relevant to the synthetic benchmark analysis.
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Unit
MMM.PHATE
Unit O
PCA2
+.SNE
Untt
Unit (8)
MM-PHATE
PCA
I-SNE
Epoch
thopt. n=3)
1.75
Readout aclivation
1.50
1.25
1.00
0.75
0.50
0.25
1000
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7000
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Global time (epoch T
Figure A.17: Hopf clean vs. Hopf warped (warp-only control): 2-D unit-level organization and average readout traces. Top: embeddings colored by unit index. Bottom: average readout traces for the clean and warped Hopf groups across global training time. These panels complete the warp-only control and support the claim that MM-PHATE is not strictly warp-invariant, but still retains the qualitative progression structure used in the downstream entropy analyses.
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Taken together, these three synthetic scenarios provide controlled qualitative evidence that MM-PHATE (1) distinguish latent dynamics-famii differences, (ii) retain interpretable family-level distinction under smooth state-space warps, and (iii) preserve progression structure even when the same latent dynamics are observed through different coordinates. We emphasize again that these are qualitative synthetic stress tests and do not. constitute a formal invariance guaranlee under arbitrary smooth equivalences.
A.5.4 Hopf anchor benchmark and entropy validation
Hopf anchor construction (dynamic vs. static Hopf groups). The Hopf anchor benchmark used for the main-text geometric and entropy validation is distinct from the symmetric 3+3 scenario suite Section A.5.2 Here, all units are generated from the same supercritical Hopf system, but with two different y-schedules: dynamic (swept-j)) group with units, for which μ linearly swept from -1 to across epochs, and a static (fixed-p) control group with units, for which μ -1 all epochs. Thus, the dynamic group crosses the bifurcation and develops post-bifurcation limit-cycle dynamics, whereas the static group remains in the pre-bifurcation contracting regime throughout training, Both groups use the same latent ODE family and the same readout construction (up to unit-level noise/jitter), so the key difference is whether the bifurcation is traversed.
The Hopf bifurcation experiment provides a controlled setting for evaluating how embedding methods capture latent geometry, readout distortions, and bifurcation structure.
As the dynamic group crosses the Hopf bifurcation, the oscillatory radius increases substantially. PCA, I-SNE, Isomap, LLE., and UMAP primarily encode this amplitude growth, inflating post-bifurcation trajectories and obscuring the bifurcation geometry. MM- PHATE instead collapses post-bifurcation epochs onto single coherent orbit, revealing the latent limit- cycle structure, rather than its raw magnitude. With tanh readout, conventional methods additionally show saturation effects, while MM-PHATE remains qualitatively unchanged. (Static units are limited for Isomap/LLa LLE due to scalability constraints.)
Epoch-colored embeddings (Figs. A.18
A.24
100
80
MM 2
PCA
TSNE
160
PHATE
Epoch
40
20
2 2
2
UMAP 2
ISOMAP
UE
UMAP
Figure A.18: Embeddings the Hopf RNN traces colored by epoch using MM-PHATE, PCA, I-SNE, Isomap, LLE, and UMAP (left to right, top to bottom). Note: For Isomap and LLE, we were wnable to include sufficiently many static wnits due to poor scalability and subsampling constraints.
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All units read out the same underlying Hopf dynam- ies but acquire subtle unit-specific variations after the bifurcation. MM-PHATR, more clearly resolves these differences, separating individual units along the shared manifold. PCA, t-SNE, UMAP, Isomap, and LLE in- stead collapse unit trajectories together, both with and without the tanh nonlincarity. They show only minor deviations from injected noise and fail to caplure the structured unit-level dynamical variation, (Isomap/LLB omit some static units as above.)
Unit-colored embeddings (Figs. A.19 A.25
NMA PHA-PHATE
TSNE 2
giden Unit
HM-PHATE 1
PCA
TSNE
H
2
ISOMAP2
LLE
UNAP 2
ISOMAP
LEE
UMAP 1
Figure A.19: Embeddings of the Hopf RNN traces colored by hidden-unit identity across six methods. Note: For Isomap and LLE, were unable to include sufficiently many static units due to poor scalobility and subsampling construints.
Singl--dynamc--uit zoom (Fig. A.20). MM-PHATE organizes the evolution of a single dynamic unit across the entire sweep: contraction for P < O, unfolding geometry near zo O, and a clean limit cycle for p > 0, with the magnified view showing a smooth circular orbit.
A
100
B
C
D
2
80
Epoch
40
IONHUATE
2
HMPHATE
MMMPHATE
MM PHATE
HICPHATE
Figure A.20: Zoomed-in views of the MM-PHATE embedding for single dynamic hidden unit across the Hopf bifurcation. (1) Full trajectory across all epochs, showing the global transition from the pre-bifurcation contraction to the post-bifurcation limit cycle (colored by epochs). (2) An early epoch with μ O, where the trajectory collapses toward the fixed point with minimal rotational structure (colored by timo-steps). (3) An epoch near the bifurcation point (μ ~ 0), where the geometry begins to unfold and organization emerges (4) late epoch with > 0, where the dynamics form a clear and stable limit cycle in the embedding (colored by time-steps). (5) magnified view of a single post-bifureation limit cycle, illustrating the smooth and consistent circular structure recovered by MM-PHATE (colored by timo-steps).
(colored by time-steps).
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Group-colored embeddings (Fig. A.21). Before bifurcation, dynamic units form diffuse cloud re- ilecting heterogeneous transients en route to the emerging limit cycle, whereas static units (constant / < 0) repeatedly collapse to the same fixed point and form a compact cluster. MM-PHATE cleanly differentiates the two groups and reveals their distinct trajectory geometrics across the Hopf bifurcation. Before bifurcation (M < 0), the dynamic units form a broad cloud, reflecting noise-dominated dynamics near the collapsing fixed point and the fact that these units are progressing toward the emergent limit cycle. contrast, the static units (whose j remains fixed < 0 across all epochs) accumulate into denser, more compact structure, since their temporal evolution is repeatedly drawn toward the same contracting fixed point. Post-biffuccatonn MM- PHATE continues to separate individual dynamic-unit trajectories, demonstrating that the method captures the full temporal progression of each unit rather than collapsing them into single manifold. In contrast, the other methods are dominated by amplitude changes and small stochastic differences between units: they produce concentric, trumpet-shaped loops in which trajectories from different units largely overlap, with only subtle separations that primarily reflect noise rather than genuincly distinct dynamical orbits.
Dynamic
MBA PHATE
PCA
TSNE
MM MMM-PHATE
PCA
TSNE
Group
Static
ISOMAP
2
UNAP 2
ISOMAP
UMAP
Figure A.21: Embeddings of the Hopf RNN colored by group identity (dynamic vs. static hidden units). Nole: For Isomap and LLE, we were unable to include sufficienlly many static unils due to poor scalability and subsampling constraints.
Intra-step entropy (Fig. A.22). Latent intra-step entropy (computed across samples) shows low entropy for static samples and sharp rise for dynamic samples near the bifurcation. The readout inverts this pattern due to radius-dominated projection. Among the evaluated embeddings in this benchmark, MM-PHATE) is the only one that qualitatively reproduces the lested latent entropy structure; PCA, L-SNE, and UMAP instead mirror the readout, indicating their sensitivity to activation magnitudes rather than dynamical organization.
Inter-step entropy (Fig. A.23). Latent inter-step entropy exhibits consistently low entropy for static samples and a clear low-to-high high transition for dynamic samples μ crosses zero. The readout again shows the opposite pattern. MM-PHATE recovers the latent trend: static units remain low entropy, and dynamic units shift from high entropy in the noise-dominated regime (μ < 0) to low entropy in the limit-cycle regime (n > 0). t-SNE captures part of this transition but spreads it broadly across epochs, while PCA and UMAP mostly follow the readout's magnitude-driven structure.
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Latent intra-step (static dynamic)
Readout intra entropy (static dynamic)
-12
20
20
-14
-6.0
0-
0-
step entropy vs dynamic)
PCA intra-step entropy (static vs dynamic)
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20-
665
TSNE intra-step entropy (static dynamic)
UMAP intra-step (static dynamic)
80
-4.
0-
16
112
time (static dynamic)
time step (static dynamici
Figure A.22: Intra-step entropy analysis across the Hopf bifurcation. The first column shows ground-truth quantities: (top-left) latent intra-step entropy computed across samples (the latent space has no notion of hidden units), and (top-right) readout. intra-step entropy computed across hidden units. The remaining columns show intra-step entropy computod from the MM-PHATE, PCA, t-SNE, and UMAP embeddings. MM-PHATE most. closely reproduces the latent-space entropy dynamics: preserves the consistently low entropy of the static group and captures the sharp transition in the dynamic group from low to high entropy approaches the bifurcation point. By contrast, PCA, t-SNE, and UMAP fail to reflect the organization present in the latent dynamics, yielding high entropy for the static units and persistently low entropy for the dynamic units mirerrrii the readout-level distortions rather than the underlying system dynamics.
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Latent inter-step entropy (static dynamic)
Readout inter-step (static dynamic)
20
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index (static dynamic)
unit index (static dynamicc)
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PCA inter-step entropy (static vs dynamic)
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(1000
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unit index (static dynamic)
unit index (static |dnamicc)
TSNE inter-step entropy (static dynamic)
UMAP inter-step entropy (static vs dynamic)
-55
-.0
unit index (static dynamic)
unit index (static | dynamicc
Figure A.23: Inter-step entropy across latent space (top-lcft), readout space (top-right), and four embedding methods (MM-PHATE, PCA, I-SNE, UMAP; columns 2-3). In the latent ground truth (lop=left), the 2-axis indexes samples, whereas in the readout and all embedding methods the r-axis indexes hidden units. Latent trajectories exhibit consistently low entropy for the static group, and clear transition in the dynamic group from low entropy for μ 0 to high entropy for μ 0. The readout shows the opposite static dynamic pattern, with high entropy for static units and low-to-high transition for dynamic units, driven by the radius-dominated projection. MM-PHATE is the only embedding that qualitatively recovers the latent structure: static units remain low-entropy across epochs, and dynamic units transition from high entropy in the noise-dominated regime (μ < 0) to low entropy in the limit-cycle regime (μ > 0), indicating that MM- PHATE embods units according to their full temporal trajoctorics rather than instantancous magnitude. PCA and UMAP largely mirror the readout's magnitude-based structure, maintaining high entropy for static units and low-to-high transition for dynamic units. t-SNE partially captures a high-to-low transition in the dynamic group but spreads this transition across many epochs. In contrast, MM-PHATE produces a sharp and well-localized transition around 2 0 and preserves the distinction between static and dynamic units, faithfully reflecting the underlying bifurcation geometry.
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80
M4A 2
TSHE
60
MM PHATE
PCA
TSNEE
Epoch
40
20
ISOMA
uE
URAP
ISOMAP 1
UNAPP
Figure A.24: Embeddings of the Hopf RNN traces with tanh nonlinearity applied to the readout, colored by epoch using MM-PHATE, PCA, t-SNE, Isomap, LLE, and UMAP. The saturation induced by the tanh compresses the amplitudes of late-epoch trajectories, causing PCA, t-SNE, Isomap, LLE, and UMAP exhibit visibly capped or flattened embeddings that primarily reflect this value saturation rather than the underlying dynamical progression. MM-PHATE, however, remains qualitatively unchanged: continues to collapse post- bifuuctffcctionn trajectorics onto single coherent orbit and preserves the bifurcation geometry despite the nonlinear distortion of the readout. Although the global appearance of all embeddings remains similar to the non-tanh case, these results highlight that conventional methods are sensitive to the absolute scale of activations, whereas MM-PHATE retains robustness to such transformations. Note: Isomap and LLE connol include sufficient static wiils due to poor sculability and subsampling construints.
MM P
TSME
Hidden Unit
MM.RNATE
PCA
TSNE
3
2
1
ISOMAP2
2
USAP
ISOMAP
LLE
UMAP
Embeddings the Hopf RNN traces with a tanh readout nonlinearity, colored by hidden-unit identity. Despite the saturation effects introduced by the tanh, MM-PHATE continucs to cleanly separate the trajectories of individual dynamic units after the bifurcation, reflecting its ability to organize samples according to their underlying dynamical evolution rather than their raw activation magnitudes. In contrast, PCA, t-SNE, UMAP, Isomap, and LLE mix the unit trajectories together, with embeddings largely shaped by value saturation rather than dynamical structure. As in the epoch-colored plots, Isomap and LLE omit some static wnits due to scalabilily and subsampling constraints.
Figure A.25:
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A.6 Neighborhood Preservation Analysis
We assess how faithfully each embedding preserves the local geometric relationships present in the activation tensor T R(xa)xmxX by computing intra-step and inter-step neighborhood preservation. Each time-epoch pair (T, w) corresponds to intra-step slice Tr R=xp containing the activations of all hidden units across p samples.
Common evaluation grid. Because different methods operate on different subsampled sets of (r, w) pairs (c.g.. UMAP and LLE require subsampling, Isomap requires even more aggressive subsampling), we evaluate methods on the inlersection of available slices. Isomap is excluded from the main comparison due to its extremely small intersection set under feasible subsampling.
Intra-step neighborhood presorvation. For each slice (T, and hidden unit i, compute its è nearest neighbors among (Tr hm in the z-scored activation space and among (Dr.w in the embedding, The intra-step preservation score is the average fraction of overlapping neighbors across all slices and units:
| (T,w, w. Ne (T, w. i)
NP intro-step Erg
k
Inter-step neighhorhood preservation. and epochs,
For each hidden unit i, we consider its trajectirr across time
(T, (T)T))))(w)))
and compute the nearest neighbors of each slice in both the original and embedded spaces (excluding self-neighbors). The inter-step score is the average neighbor overlap across units and slices:
[ (1: (, w) N femb (i; T, T,W)
NP inter-step Eirw
k
Reported results. For each method and & E {5, 10, 15, 20, 40), we report
(NPitra-step:: NP (iuter-step)
computed on the exact same intersection of (T, w) pairs, ensuring that differences reflect the embedding quality rather than differences in sampling or data coverage.
Intra-step preservation
Inter-step preservation
1.0
preservation
Neighborhood preservation
Neighborhood
0.4
Isomap
Isomap
MIM-PHATE
MM-PHHATE
PCA
PCA
t-SNE
E-SNE
5
10
15
20
40
10
15
20
40
k (number neighbors)
(number of neighbors)
Figure A.26: Area2Bump: Neighborhood preservation of visualization methods including Isomap.
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Table Neighborhood preservation of visualization methods (Without Isomap).
MM-PHATE
PCA
t-SNE
UMAP
LLE
Intra-slep
0.649
0.475
0.515
0.300
0.299
Inter-step
0.671
0.519
0.574
0.657
0.703
10
Intra-step
0.789
0.717
0.741
0.546
0.546
10
Inter-step 0.672
0.473
0.544
0.659
0.647
15
Intra-step
0.859
0.860
0.863
0.814
0.812
15
Inter-step
0.667
0.466
0.528
0.630
0.599
20
Intra-step
1.000
1.000
1.000
1.000
1.000
20
Inter-step
0.660
0.471
0.517
0.594
0.560
40
Intra-step
1.000
1.000
1.000
1.000
1.000
40
Inter-step
0.634
0.511
0.534
0.476
0.426
Intra-step preservation
Inter-step preservation
1.0
1.0
Neighborhood preservation
Neighborhood preservation
MM- HIN-PHATE
PCA
0.8
0.8
L-SNE
0.6
0.6
0.4
0.4
LLE
0.2
PCA
0.2
UMAP
0.0
0.0
5
20
40
10
15
k (number of neighbors)
(number of neighbors)
Figure A.27: Arca2Bump: Neighborhood preservation without Isomap
A.7 Time-resolved linear probing analysis
To quantify how much task-relevant information finearly decodable from recurrent representations dif- ferent positions in the sequence, we performed time-resolved linear probe analysis on the hidden states h, of the recurrent network R. Let X denote the supervised dataset with labels L, and let be the total number of time-steps in the sequence. At collection of training epochs т E {1...., n (subsampled for efficiency), we evaluated the trained network on the fixed training and test splits and extracted the sequence of hidden states (he Ji= for each input example.
Per-time-step linear probes. For each probed epoch and each time-step w E {l....,s) s}, we formed a feature matrix by collecting the hidden state that time-step across examples, yielding H(Fw) E Nxm. where m the number of hidden units and N is the number of evaluated examples. We then fit multinomial logistic regression (linear classifier) to predict the class labels L from H(ww) using the training split, and evaluated the resulting probe on both the training and test splits. This yields time-resolved probe accuracies (r, w) and "r, which quantify the extent to which label information is lincarly accossible from the representation at a specific time-step and epoch.
Heatmap summaries over epoch and time. We visualize lestest (T,w) and train (T,w) w) as heatmaps (Fig. A.28 with axes (epoch r) x (time-step w). These summaries reveal where in the sequence and when during training label information bocomes lincarly scparable in the hidden state. In our experiments, later time-steps exhibit steady increase in test decodability over training and reach higher, more stable platean, whereas early time-steps show more transient test. decodability that peaks mid-training and later degrades.
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199
199
0.56
154
0.45
Probe test accuracy
154
Probe train accuracy
0..4
109
109
Epoch
64
0.30
Epoch
64
0.40
27
27
18
0.15
18
0.32
9
0.24
99
199 299 399 499 599
0 99 199 299 399 499 599
Time step
Time step
Figure A.28: Heatmaps showing linear probe accuracies across training epochs and time-steps. Left: Probe test accuracy, illustrating the model's performance on the test set for each time-step at different epochs. Right: Probe train accuracy, showing the model's performance on the training set across epochs and time- steps. Both heatmaps visualize the evolution of probe accuracy over time, with the color intensity repre- senting accuracy values.
Train-test gap by time segment
0.150
Train - test accuracy
0.125
0.100
0.075
0.050
0.025
Early gap (train test)
Late gap (train test)
0.000
100
150
200
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Figure A.29: Train-test accuracy gap over training epochs, separated into early and late time segments. The plot compares the difference between training and testing accuracies for early and late time steps, illustrating how the gap evolves throughout the training process. Early time steps are represented by circles, while late time steps shown with squares. A larger gap typically indicates overfitting, while smaller gap suggests more generalization.
Carly/late aggregation and train test gap. To align the probe results with the early-vs-late compar- isons used elsewhere in the paper, we further aggregate probe accuracies over "early" and "late" segment of the sequence (defined by fixed boundary time-step 350 based on divergence of intra-step entropy trends). For cach epoch we compute mcan probe accuracics within each segment for both training and test splits, and report the corresponding generalization gaps (Fig. A.29h:
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We visualize these quantities in Fig. A.29 to highlight when and where overfitting emerges in the sequence. Consistent. with the main-text findings, the early-segment gap increases substantially after mid-training,
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coinciding with the deterioration of early time-step test probe accuracy, whereas the late segment maintains higher test decodability with a smaller train-lest, gap.
Overall, the time-resolved probe analysis provides complementary validation that the representational patterns observed in the MM-PHATE entropy analyses track task-relevant information: label signal becomes increasingly decodable at later time-steps, while early time-steps eventually exhibit higher variability that does not translate into improved generalization.
A.8 Time-step ablation analysis
To assess how the trained recurrent classifier R uses information across the sequence, we performed a time- step ablation analysis in which portions of the input sequence were masked at evaluation lime while keeping the network weights W, b fixed. Let X denote the evaluation dataset with labels L, and let be the total number of time steps. For set of training epochs т (subsampled consistently with the entropy and probing analyses), evaluated classification accuracy under three input conditions:
1. Full sequence (intact): the original input sequence is provided to the network.
Early-only: time steps w > we are zeroed (late segment ablated), preserving only the early portion of the sequence.
3. Late-only: time steps w & wo are zeroed (early segment ablated), preserving only the late portion of the sequence.
Here wy (equals 350 in the example) is a fixed boundary time step defining the early/late split (the same split used to summarize probe results). For each epoch т, we load the corresponding trained parameters and compute accuracy on both the training and test splits under each masking condition. Because masking is applied only at evaluation time, differences in accuracy directly reflect the extent to which the ezisting trained decision rule relies on early versus late parts of the sequence, rather than changes in training dynamics.
Figure A.30 reports the training-set accuracy across epochs for the intact model and the two ablated settings. Consistent with the main-text interpretation, carly-only performance rises during early training but does not sustain the intact model's final performance, while late-only performance improves gradually over training yet remains below the intact sequence. These results not only indicate that final performance depends on integrating information across the full sequence but also provide insight into how the network's s reliance different sequence segments evolves during training. Early time-steps provide shortcuts that are initially leveraged by the classifier, but as training progresses, the reliance shifts toward later time-steps, culminating in a model that integrates information across the entire sequence for robust generalization.
Taken together, these results indicate that although both segments can support nontrivial classification when isolated, the strongest performance depends on coherent integration across the full sequence, and the network's reliance gradually migrates from early to later time-steps as training continues.
A.9 Time-resolved mutual information analysis
To measure the total task-relevant label information present in the recurrent representation, independent of any particular readout, we computed time-resolved mutual information (MT) between class labels L and the hidden state h4 of the recurrent network R. Let denote the total number of time steps and m the number of hidden units. At a set of probed training epochs 7 (subsampled consistently with the entropy and probing analyses), we extracted the hidden-state tensor on the training set and estimated MI at each time step and hidden unit.
Per-time-step, per-unit MI. For each epoch 7 and time step w E {l....,s), we considered the hidden state voctor across examples and treated each unit as a continuous feature. We estimated the mutual information between L and each unit activity at (7,w) using a nonparametric estimator for continuous features, producing a time-ume unit MI array. We then summarized the overall label information each time
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Figure A.30: Time-step ablation on the training split across training epochs. We compare the intact model (full sequence) to early-only evaluation (late time steps zeroed) and late-only evaluation (carly time steps zeroed). The intact model consistently achieves higher accuracy than either ablated condition, indicating that final performance relies on integrating information across the full sequence rather than any single segment alone. The gradual shift from early reliance to full-sequence integration reveals the network's evolving strategy for learning and generalization.
step by averaging MI across units, yielding time-resolved MI curve [=) that reflects how much label information is present in b at epoch regardless of whether the trained classifier exploits it.
Heatmap summary across epochs and time steps. Figure A.31 visualizes the mean-over units MI L (w) as a heatmap with axes (epoch r) x (time step w). This representation highlights how label information accumulates and redistributes across the sequence during training. In our experiments, MI increases over training and is consistently higher at later time steps, corroborating the probe and ablation results that later portions of the sequence carry more task-relevant information in the learned hidden dynamics (Fig. 6
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Figure A.31: Timeeresolved mutual information between labels and the recurrent hidden state has sum- marized by averaging MI across hidden units (m) at cach epoch and time step Warmer colors indicate greater label information present in the representation at that time and training stage. The heatmap shows increasing label information over training, with consistently higher MI at later time steps.
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A.10 Random Label Analysis
A.10.1
Experimental setup
To probe how our MM-PHATE and entropy summaries behave in an extreme overfitting regime, we per- formed a random-label experiment inspired by Fischer (2020). The idea is progressively destroy the correspondence between inputs and labels and ask whether the resulting representational changes are re- flected in MM-PHATE geometry and intra-/inter-step entropy.
We trained multiple 1-layer LSTM models with 100 hidden units on the Area2Bump dataset under different label-shuffing conditions. For each condition, a specified number of classos had their labels randomly reas- signed while the remaining classes retained their true labels. For each shuffle level, we drew 10 independent random plass-permutationn configurations. All models were trained with the same hyperparameters (learning rate 10-4 batch size and optimizer as in Section 5.2), and most runs reached (near- -zero training loss within 200 epochs, while training shuffled labels is known to severely degrade test performance.
We focused on the final training epoch and extracted hidden activations across all 600 time-steps. On this epoch we computed: (i) MM-PHATE embeddings of the hidden units across time and epoch, (ii) intra-step and inter-step entropies in the MM-PHATE space (as defined in Section 5.2). and (iii) summary statistics and distributional distances of the intra-step entropy across time-steps, using a clean-label model as a reference.
Specifically, for each shuffle level we summarized the intra-step entropy at the last epoch by: (i) maximum entropy across time-steps, (ii) mean entropy across time-steps, (iii) entropy al the last. time-step, and (iv) variance of entropy across time-steps. We then compared the entropy distributions over time-steps between shuffle levels using Jensen-Shannon divergence (JSD) and total variation distance (TVD), both computed relative to a reference model trained on correct labels.
A.10.2 Metrics and analysis
Intra-step entropy. As in Section 5.2 intra-step entropy is computed at each time-step by applying KDE-based estimator to the cloud of embedded hidden units at that time-step and epoch. It is therefore a measure of geometric dispersion across units in the MM-PHATE embedding, not a direct estimate of information-theoreti. entropy of neural responses. In this experiment, we use its summary statistics as a compact way to track how the spread of unit representations across time-steps changes as label noise increases.
Inter-step entropy. Inter-step entropy is computed per unit and epoch, capturing how dispersed unit's embedded trajectory is across time-steps. Larger inter-step entropy indicates that the unit differentiates more strongly across time within an epoch; smaller values indicate more temporally homogeneous activity. Here we examine how these unit-level temporal profiles change with label shuffling.
Distributional distances (JSD, TVD). To quantify how much the overall entropy profile the last epoch deviates from the clean-label regime, we treat the intra-step entropy values across time-steps ono-dimensional empirical distribution and compute: (i) the Vensen-Shannon divergence between this distribution and that of the clean-label model, and (ii) the corresponding total variation distance. These distances do not by themselves identify "good" or "bad" representations, but they provide simple way to measure how strongly the entropy profile shifts as labels are progressively randomized.
A.10.3 Results
MM-PHATE geometry. With no minimal label shuffling, MM-PHATE produces embeddings that show structured trajectorics across epochs and relatively organized arrangement of hidden units over time (Fig. A.32 As more classes are shuffled, these structures become progressively less differentiated: trajectories appear more tangled across epochs and units, and the embedding shows weaker separation consistent with the loss of stable, task-aligned structure. This qualitative change not meant as fine- grained diagnostic of class structure, but demonstrates that MM-PHATE geometry sensitive to the transition from meaningful learning fitting random targets.
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Figure A.32: Random label experiment: MM-PHATE visualization of 100-unit LSTM networks trained for 200 epochs on Area2Bump training data with different numbers of shuffled classes (indicated in the legend). Each point is hidden unit at a given timo-step and epoch. Points are colored by epoch.
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Figure A.34: Random label experiment: inter-step entropy of the MM-PHATE embedding for 100-umit LSTM networks trained for 200 epochs Area2Bump, with different numbers of shuffled classes. Each curve shows the entropy trajectory of hidden unit across time-steps at different epochs.
Intra-step and inter-step entropy trends. Across shuffle levels, intra-step entropy and inter-step en- tropy exhibit systematic changes (Figs. A.33 A.34). In the clean-label condition, intra-step entropy over training shows the structured temporal patterns described Scction 5.2 As more classes are shuffled, these patterns altenualed and the entropy trajectories across epochs and time-steps become more homogeneous. At the final epoch, summary statistics of the intra-step entropy (maximum, mean, last time-step value, and variance) decrease with incrcasing label shuffling (Fig. A.35 indicating that the embedding-spee spread of unit representations across time-steps becomes more concentrated and less temporally differentiated under heavy label noise.
Inter-step entropy traces also change with shufiling. With true labels, units exhibit distinct temporal profiles over time-steps; as shuffling increases, many inter-step entropy trajectories become larger and less structured across epochs (Fig. AA.44,, consistent with units varying over time in ways that are less clearly aligned with the training phases seen in the clean-label regime. We interpret these trends evidence that the temporal organization captured by MM-PHATE entropy summaries is disrupted when the network is forced to fit inconsistent labels.
Distributional distances. Finally, JSD and TVD between the intra-step entropy distributions (over time- steps) at the last epoch and the corresponding clean-label reference increase more classes are shuffled (Fig. A.36 A similar trend is observed when computing these distances on the embedded point clouds themselves. These distances capture, in a single scalar per condition, how far the entropy profile has moved away from the structure observed in the clcan-label model.
Overall, the randon-label experiment serves as stress test: when we deliberately destroy the input label relationship, MM-PHATE geometry and the associated entropy summaries change in systematic way, While do not claim these mcasures uniquely quantify "reprosentation quality," " their sensitivity to this controlled overfitting regime supports their use as indicators of regime changes and loss of lask-aligned structure in the main experiments.
A.11 Area2Bump with GRU
Here is the same analysis as section 5.2 using GRU (Fig. same.
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From these figures, is evident that PCA and t-SNE present similar visualizations the hidden dynamics across different network architectures, while MM-PHATE distinctly captures the unique learning behaviors PCA displays increasing intra-step entropy even after model
of each model. Consistent with Section 5.2
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accuracy has plateaued, and t-SNE produces noisy visualization. In contrast, MM-PHATE more clearly aligns its transitions well with the learning curve. Notably, the GRU model's representation appears more compact and organized compared to the LSTM model, potentially reflecting its superior performance and reduced overfitting.
A.12
Area2Bump with Vanilla RNN
Here is the same analysis as section 5.2 using vanilla RNN (Fig. A.40
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From these figures, it is evident that regardless of the network architectures, PCA exhibits a revolving pattern with overly smooth transitions across epochs, while t-SNE produces noisy visualization. In contrast, the MM-PHATE visualization reveals that the Vanilla RNN displays more chaotic pattern compared to the LSTM and GRU models, which is likely associated with its reduced performance and incrcased overfitting. Furthermore, the intra-step entropies of MM-PHATE show reduced variation across time-steps, indicating that the model struggles to process the input data effectively to generate meaningful representations.
A.13 Area2Bump with LSTM of Various Sizes
Here we repeat the same MM-PHATE analysis as in section 5.2 using LSTM of various sizes (Fig. A.44 A.45
A.46J. Other parameters were kept the same. These results demonstrate that MM-PHATE consistently captures smooth yet distinct transitions across epochs and lime-steps, regardless of the LSTM network size. The intra- and inter-step entropy analyses further reveal that these transitions closely correlate with performance changes throughout training. Specifically, we observe a general increase in intra-step entropy as models begin to overfit, suggesting that the networks increasingly memorize input information. In contrast, inter-step entropy shows significant decline overfitting progresses, reflecting loss of sensitivity to input
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divergence (JSD) and total variation distance
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Figure A.37: Area2Bump GRU: Visualization of 20-unit GRU network trained for 200 epochs. Each point represents hidden unit at specifie time-step given epoch throughout the entire training process. The visualizations generated using MM-PHATE, PCA, and t-SNE, from left to right, respectively. Points are colored based on epoch (top row) or time-step (bottom row)
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Figure A.39: Area2Bump GRU: Inter-step entropy of all hidden units in embedding space of the Area2Bump model each time-step in each epoch, compared to training and accuracies (top) and losses (bottom). From left to right, the dimensionality reduction metrics used are MM-PHATE, PCA, and t-SNE.
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Figure A.41: Area2Bump Vanilla: Intra-step entropy of all hidden units in embedding space at each time- step in each epoch, compared to training and validation accuracy (top) and losses (bottom), comparing embeddings of MM-PHATE, PCA, and t-SNE.
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Figure A.42: Area2Bump Vanilla: Inter-step entropy of all hidden units in embedding space of the Arca2Bump model at each timc-stop in cach epoch, compared to training and accuracics (top) and losses (bottom). From left to right, the dimensionality reduction metrics used are MM-PHATE, PCA, and I-SNE.
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Figure A.43: Area2Bump: Inter-step entropy of all hidden units in embedding space of the Area2Bump model at each time-step in each epoch, compared to training and accuracies (top) and losses (bottom). From left to right, the dimensionality reduction metrics used are MM-PHATE, PCA, t-SNE, Tsomap, L.I., and UMAP.
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Figure A.46: Area2Bump LSTM: Inter-step entropy of all hidden units in MM-PHATE embedding space of the Area2Bump model at each time-step for each unit in each epoch, compared to accuracies (top) and losses (bottom). Network sizes range from 10 to 50 (left to right).
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A.14 Quantitative cumporal-ggemettre metrics for HAR MM-PHATE embeddings
33d reduce reliance on visual inspec- tion alone, we computed two epoch-wise quantitative metrics directly from the 3D MM-PHATE embedding of the HAR LSTM hidden states: (i) signed_ wlww_wlllgnaaaaaeeaeell cetpantddeeerddddreeee which measures the direction of within-sequence progression at each epoch, and (ii) epoch_to_ _epe_ttt_eppheeeeeeeeeeee__eeeee saag___hh_______e _magnitude, which measures the amount of temporul-geomerri reorganization between consecutive sampled epochs.
To complement the qualitative MM-PHATE visualizations in Section 5.3
Epoch-wise temporal centroid trajectories (unit-balanced). Let Zr,,l E R3 denote the 3D MM- PHATE point associated with sampled epoch т, intrinsic time-step E {1,....8), and hidden unit i e (l....,m) m). whenever that tuple is present in the subsampled embedding. In our HAR pipcline, each observed (T,w,)) tuple contributes at most one MM-PHATE point; subsampling removes some tuples but does not create duplicate tuples.
Bocause some (T, (r, i) tuples may be missing after subsampling, we define the set of hidden units present epoch 7 and time-step w as
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2, (ub) 7,
(12)
The reference axis á is estimated from the first principal component of (g+} across epochs with sufficient observed time-steps.
Next, define consecutive centroid increments along the within-epoch trajectory:
A
(ub)
(ub)
k k= L Kr - 1.
(13)
7,4241
7,w6
We define the figned_flow www.gnmeet_ceetrtteeeooooo
_K.-1 (A à)
A, =
(14)
k==1 A
which is bounded in []1,1] (up to numerical error). Positive values indicate that the within-sequence trajectory predominantly progresses along â, whereas negative values indicate progression in the opposite direction. This yields an epoch-wise directional summary without pre-specifying training regimes.
Epoch-to-epoch change magnitude (temporal-geometric reorganization). A change in temporal dynamics may appear not only sign flip in Eq, equation [4] but also changes in temporal ordering, trajectory extent, or spread. To capture this more broadly, we compute epoch-wise temporal-geometric signature vector
A
P+ q.
(15)
l
&
where:
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A, is the signed flow alignment in Eq. equation 14
Pr is the Spearman correlation between pairwise centroid distances | _T.aa
(ub)
and time-step
lags v/ (temporal ordering strength),
+ - | is the within-epoch centroid trajectory path length (trajectory extent), & is the within-epoch centroid spread across time-steps, computed as the mean coordinate-wise variance of (ub) over observed w.
Let TI < 72
TR denote the sampled epochs used in the embedding/ faalinng/mmee /metric computation (which may be subsampled and therefore non-consecutive). After z-scoring each component of qr, across ,,...,R R. the epoch_ to_ _e__ph change_manneeee caaelemmannttii tude is defined as
C, | 4r. 1:.
T=
R
(16)
and is plotted at the midpoint (T, + 1/2 This formulation avoids imposing continuity assumptions when epoch labels are subsampled.
These metrics quantitatively support the qualitative MM-PHATE observations reported in iention 5.3 In particular, signed_ flow aligmment centroid (A,) getngent_ccettoiddd increases from negative to positive around epoch 7 (Fig. A.47 zoomed panel), indicating a sign change in the dominant direction within-sequence progression the MM-PHATE embedding. This sigu change coincides with pronounced spike in epoch_ tecchtt______eeee tt__tt_________
Quantitative support for the HAR transition.
magnitude (C,)) and sharp increase training/validaan /alidation accuracy, consistent. with early training-phase reorganization rather than a purely gradual drift. The corresponding shift in temporal dynamic direction is also visible in the original 3D MM- PHATE embedding (Fig. A.49,, where the direction of time-step progression changes across early epochs.
Alignment with training dynamics over the full trajectory. Across the full training run, large fluctuations in C, and changes in A, align with changes model performance (accuracy and loss; Figs. A.47 A.48). This consistent alignment supports the interpretation that MM-PHATE is capturing meaningful cemporal-gcomottie structure in the evolving hidden-state dynamics, rather than arbitrary low-dimensional distortion.
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MM-THATEE geometrii metric
Train accuracy
Validation accuracy
Signed flow alignment (full training)
Signed flow alignment (epochs 1-50)
0.5
0.20
Signed flow aligmment centroid
1.0
sigmed flow aligmment centroid
1.0
0.4
0.15
0.8
0.8
0.3
Accuracy
0.10
Accuracy
0.6
0.66
0.2
0.4
0.05
0.4
0.1
0.2
0.00
0.2
0.0
0.0
-0.05
0.0
200
400
600
800
1000
10
20
40
50
Epoch
Epoch
Epoch-to-epoch change magnitude (full training)
Epoch-to-eue -oeechch change magnitude (epochs 1-50)
epoch to epoch change magnitude
1.0
epoch to epoch change magnitude
2.0
Accuracy
2.0
Accuracy
1.5
1.5
0.4
0.4
1.0
1.0
0.5
0.2
0.5
0.2
0.0
0.0
0.0
0.0
200
400
600
800
1000
10
20
30
40
50
Epoch
Epoch
Figure A.47: Quantitative geoomerree metrics on HAR MM-PHATE embeddings, compared with training and validation accuracy. Top: signed_ 2lox_alignment centroid (A,) over training (left: full
training: right: zoom to carly epochs). Bottom: epoch_ to seocc_tt_epoch____ change_magnin
www.mmaniiutu tude (C,) (left: full
training; right: zoom to early epochs). The early sign change in A, (around epoch 7) coincides with spike in C, and sharp increase in accuracy.
65

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Published in Transactions on Machine Learning Research (02/2026)
MM-PHATE gerpmeta--gommtriicc metric
Train loss
Validation Joss
Signed flow alignment (full training)
Signed flow alignment (epochs 1-50)
0.5
1.8
0.20
1.8
Signed flow alignment centroid
signed flow alignment, centroid
0.4
0.15
1.4
1.2
0.10
1.4
1.0
Loss
Loss
0.2
0.8
0.05
1.2
0.1
0.6
0.4
0.00
1.0
0.2
-0.05
200
400
600
1000
10
40
50
Epoch
Epoch
Epoch-to-epoch change magnitude (full training)
Epoch to-ehhhh change magnitude (epochs 1-50)
epoch to epoch. change magnitude
1.8
epoch to epoch change magnitude
1.8
1.6
2.5
1.4
1.2
2.0
2.0
1.4
Loss
1.0
1.5
1.5
0.8
1.2
1.0
1.0
0.6
0.5
0.4
0.5
0.0
0.2
0.0
200
400
600
800
1000
10
20
30
40
50
Epoch
Epoch
Figure A.48: Same quantitative temporal-geometricc metrics as Fig. A.47 but compared with training and validation loss. Large fluctuations in the MM-PHATE temporal-goometric metries align with changes loss over training, supporting that the embedding captures training-relevanl temporal reorganization in the hidden-state dynamics.
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MM-PHATE
50
MM-PHATE (epochs 1-50)
120
40
100
30
80
Time-step
Epoch
60
20
40
10
20
Figure A.49: Early-epoch HAR MM-PHATE 3D embeddings (first 50 epochs), shown from fixed viewpoint. Left: colored by epoch т. Right: colored by intrinsic time-step The time-step-colored view makes the early shift in temporal progression direction visible, consistent with the sign change in signed_ feoowddflww___ wwwplllooowlllleee wewiiinnnttceerreee (A,).
Time-step Index
Time-stop Index
entropy ball
1.8
1.2
stop entropy
L.1
02
Intra-4tep entrropy
3.1
1.6
2.0
L,00s
0.9
0.4
0.9
3.2
0.8
0.0
0.2
1.0
0.7
0.7
0.0
Epoch
Figure A.50: Early-epoch HAR MM-PHATE intra-step entropy (first 50 epochs). Left:
compared
accuracy. Right: compared loss. The intra-step entropy changes drastically around epoch 7, aligning the sign change in signed flow wetgggnnt_ceettrree (A,) and performance increase.
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Published in Transactions on Machine Learning Research (02/2026)
B Computing Infrastructure
All but the t-SNE Arca2Bump computation was carried out a 14-core laptop running Windows 11 Home with a NVIDIA Geforce RTX 3070 Ti Laptop graphics card and 40GB of RAM. The I-SNE Area2Bump visualization was conducted on a single 95-core internal cluster running Ubuntu 18.04.6 LTS with 10 Quadro RTX 5000 graphics cards and 755GB of RAM.
C Mathematical Notations
Notation
Definition
x²
Point in high dimensional space
E
Euclidean distance matrix between all data points Affinity kernel matrix
K ((1;)
k-nearest-ncighbor distance of z;
a
Parameter controlling the decay rate
Diffusion operator
D
Diagonal matrix row sums K Transition probabilitios of diffusion process over steps Total number of epochs the nelwork is trained for
p
F
Feed-forward neural network
Total number of hidden units in the network
X
Training data, subset of II Larger dataset.
II
T
Activation tensor
Y
Input data, subset of X with equal number of samples per class
P
Number of samples in
(+)
intraslice (i,j) K(t) interslice (T, v))
Intraslice affinities between pairs of hidden units within an epoch 7 Interslice affinitics between a hidden unit and itself different epochs Intraslice bandwidth for unit in epoch T
(r,))
Fixed interslice bandwidth Index for given epoch
i, j
Index for given hidden unit
RNN hidden state at time-step t
W
RNN weights
b
RNN biases
Yt
RNN output at time-step 1
f
RNN activation function
R
Recurrent neural network
w
Index for given time-step Total number of time-steps in the RNN
8
K [TwW] intra-step
Intra-step affinities between hidden units i and at time-step in epoch
k-())
intor-stop ((T, w). (1,, v))
Inter-step affinities between hidden unit i and itself at different time-steps and epochs Intra-step bandwidth for unit i at time-step w and epoch Number of nearest neighbors Labels of
(T.w.))
k
Table 4: Notations
68


Reviews:
[{"review_id": "EzugJ2gHQX", "reviewer": "Reviewer_f4wa", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper presents MM-PHATE, an extension of M-PHATE for visualizing training dynamics of RNNs by using structured kernel also across time steps of the RNN. The paper evaluates the proposed technique on two datasets, namely Area2Bump and the Human Activity Recognition, and empirically shows using visual plots that MM-PHATE preserves community structure among units and identifies information processing and compression phases during training\n\n\nStrengths:\n- New approach for visualizing the dynamics of RNNs throughout their training\n- Results on two datasets, Area2Bump and the Human Activity Recognition, visually illustrates the benefits of the proposed approach\n\nWeaknesses:\n- Technically, this seems like a straightforward adaptation of M-PHATE, where the affinities are computed in a similar manner, just expanding the scope from across epochs and hidden unit to across epoch, hidden unit, and time step.\n- There is no sufficient quantitative evaluation and comparison between approaches, making it hard to judge the claims in the paper, in particular in comparison to the baselines, instead relying on interpretation of the plots.\n\t- In particular, contribution #2 (\"MM-PHATE preserves the community structure of hidden units by tracking their learning trajectories and capturing correlations among their activations throughout training”) is therefore not sufficiently substantiated in comparison to the baselines.\n\t- I note that the precursor work (M-PHATE) has provided some quantitative analysis and comparison of the preservation of neighborhood structure.\n- The claim in contribution #3 (\"MM-PHATE reveals phases of information processing and compression during training, consistent with the information bottleneck theory”) is not sufficiently justified:\n\t- This claim is not clearly formalized mathematically so it is hard to verify.\n\t- The experiments designed \"to quantitatively confirm that the information retained by the network is reflected in the MM-PHATE embedding” are again based on looking at plots and not based on a clear quantitative comparison to the baselines.\n\t- The random label experiments (in the appendix) are only conducted on MM-PHATE and not on the baselines and do not support the comparison between the baselines.\n- The lack of quantitative comparison, and the reliance on visual plots also restricts the experimental analysis to two datasets. It would also be nice to see analysis on different types of sequential data like, e.g., natural language data.\n- Comparison with the precursor work (M-PHATE) as a baseline is very limited and only seems to include Fig A.7. It is not clear why not all comparisons with the baselines include M-PHATE", "claims_and_evidence_yn": "No", "claims_explanation": "As noted under weaknesses above, contributions #2 and #3 are not sufficiently supported by clear and convincing evidence to show the gains of the proposed approach over the baselines, and do not include potentially relevant baselines.", "audience_yn": "Yes", "audience_explanation": "As noted under weaknesses, this proposed approach seems like a relatively straightforward adaptation of M-PHATE and the potential uses of this tool are not entirely clear. However, its ability to visualize the training dynamics likely makes it useful for researchers working with RNNs.", "requested_changes": "- The main request is to better support contributions #2 and #3 via quantitative measures and clear quantitative comparison with the baselines to show the gains of the proposed approach.\n- Provide full comparison with M-PHATE\n- It would be nice to see how such analysis works in NLP tasks (not critical)", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "7pTWlQf3kE", "reviewer": "Reviewer_8qeY", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "- RNN are widely used yet still remain a largely a black box. To alleviate this issue this paper proposes Multiway Multislice PHATE (MM-PHATE), an extension of the M-PHATE approach. The authors extension considers affinities between units at a given time step as well as between time-steps.\n- The paper demonstrates that the new method preserves hidden unity structure\n- The paper presents interesting new evidence in support ot the information bottleneck theory for neural networks.\n- The related work section summarizes the literature well.\n- The authors observe improved temporal consistency when using the new method on the Human Activity Recognition (HAR) as well as the Area2Bump data-sets. The area2bump datasets consists of neural spiking activity data from macaques.", "claims_and_evidence_yn": "Yes", "claims_explanation": "- The presented evidence supports the paper. I saw that the supplementary code sets the pseudorandom seeds values. If the authors add the exact version of the machine learning library they used, this work can be made reproducible.\n- Unfortunately the data sets used in this paper are relatively obscure.", "audience_yn": "Yes", "audience_explanation": "- Neat illustrations, the paper is relevant and well written.\n- I expect the paper's experiments on neuroscience-data will fall on fertile ground within the machine learning community.\n- Experiments on more widely known data-sets would broaden the paper's reach.", "requested_changes": "- Add a citation for the first mention of the information bottleneck theory on page 2.\n- The linear transformers are RNNs paper ( https://proceedings.mlr.press/v119/katharopoulos20a/katharopoulos20a.pdf ), could be cited to strengthen the last point in the conclusion.\n\n- It would have been interesting see the method applied on more widely know data sets like WMT'14 from ( https://arxiv.org/pdf/1409.0473 ). Although this can still happen in future work. A very quick remedy would be to include adding and memory problems from https://proceedings.mlr.press/v48/arjovsky16.pdf .\n\n- In its current form, the supplementary code is not presentable:\n    - The documentation of the supplementary code is extremely sparse, I suggest the authors improve the documentation.\n    - Furthermore the project's `requirements.txt` file does not specify the exact version of the machine learning framework used, which is important since changes i.e. to the pseudorandom number generator impact reproducibility.\n    - The code does not apply systematic containerized testing, see\n        - https://docs.pytest.org/en/stable/\n    - as well as\n        - https://nox.thea.codes/en/stable/index.html\n        - https://tox.wiki/en/4.31.0/\n    - for more information.\n    - The project code is not properly packaged, this increases the risk for code duplication in the community.\n        - Please take a look at https://packaging.python.org/en/latest/ for the final version of this code.\n    - This can still be fixed, I would consider this a minor yet very important change since replicability and reproducability are a key components of TMLR's scientific culture.", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "2iZl9VUHUI", "reviewer": "Reviewer_J7ag", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper describes a new method for visualizing the hidden dynamics of RNNs during training.\nThe method is shown to be able to identify changes in network performance, even on validation data. \nThese insights are demonstrated in two RNN training examples.\nThe proposed method generally provides more insight into the network dynamics during learning than other visualization methods.\nThe evaluation of the method is however limited in scope, often relying on a single trained RNN for a parameter/architecture set.", "claims_and_evidence_yn": "No", "claims_explanation": "## Claims with evidence\n1. \"Our analysis of the MM-PHATE embedding revealed a general increase in entropy (Fig. 3) around epoch 100, where the network begins to overfit.\"\nThis is a very interesting and useful property of MM-PHATE with clear evidence for the trained LSTM networks and a good demonstration of how other visualization methods fail to capture this. \nHowever, it is unclear that this is something specific for this example or whether this is a general feature. Including another example of overfitting would make this claim stronger.\n\n1. The method gives consistent results even for different architectures and network sizes, especially for the time-step dimension (e.g. Figure A.15). \n\n1. The clustering analysis provides some interesting insight into the internal workings of the RNN that was investigated.\n\n1. The method generally outperforms other visualization methods considered in the paper, namely PCA, t-SNE, UMAP, ISOMAP and LLE.\nFor example, in Figure 2, the structure identified by MM-PHATE is much more intuitive and interpretable than the embeddings of the other methods.\n\n\n## Claims without clear evidence\n1. It is claimed that MM-PHATE is a novel method for visualizing the evolution of RNNs’ hidden states. \nI do believe that the method is promising, but it is unclear what structures are captured/ignored by the method and how reliable the method is in different settings, see the points below.\n1. \"fluctuations in validation accuracy before epoch 30 were not reflected in the training loss curve but were detected by MM-PHATE\" but it is unclear whether this is concluded from Figure 2 or Figure 3. Having looked at both figures for a while, I am still just guessing how this might be read out from the top left subfigure from Fig 2. Perhaps having a figure in the supplementary with only the first 30 epochs could clarify how the fluctuations in validation accuracy were detected by MM-PHATE.\nLooking at Figure 3, I would have concluded that the representation in the first 30 epochs do not change, even though the loss/accuracy are changing. \n\n1. It remains unclear what exactly MM-PHATE can (and cannot) capture.\n\nFor example, why does an increase in the training accuracy correspond to the change in intra-step entropy for single units? Does this suggest that these units are responsible for overfitting?\n\nWhat kind of insight does Figure 5a provide? The figure does not seem to indicate any particular interpretable structure.\n\n1. To demonstrate the effectiveness and reliability of the method the authors should demonstrate (e.g. through correlation) that the claimed evidence of validation loss change is indeed reflected in MM-PHATE.\n\n\n1. \n\"Certain units exhibited significantly higher inter-step entropy, indicating greater sensitivity to input changes over time and a potential role in capturing temporal dependencies\".\nThis is indeed an interesting insight. \n\n\n1. Why was Dynamic Time Warping used? (Citation needed!)\nHow robust is this clustering algorithm?\nThese experiments should be repeated in different settings or instantiations to provide certainty about the reliability. \n\n1. Looking at LSTM and vanilla RNNs seems to affect the \n\nFigure A.12 seems to suggest that PCA is a better method to reflect changes in training and validation accuracy and loss than MM-PHATE as its entropy changes over time (while it seems to be constant for MM-PHATE). \n\n1. The number RNN units seems to also affect the insights provided by the method.\nFigure A.15 is less easily interpretable compared to Figure 2.\nFor example, why do later epochs evolve in two different direction along MM-PHATE 3?\nHow do the authors explain what the method is capturing?\n\n1. Overall, having different instantiations for each RNN parameter setting could clarify how robust the method is. \n\n\n\n1. Other methods that have been applied to describing RNN training:\nOstrow, M., Eisen, A., Kozachkov, L. and Fiete, I., 2023. Beyond geometry: Comparing the temporal structure of computation in neural circuits with dynamical similarity analysis. Advances in Neural Information Processing Systems, 36, pp.33824-33837.\n\nGuilhot, Q., Wójcik, M., Achterberg, J. and Costa, R.P., 2024. Dynamical similarity analysis can identify compositional dynamics developing in RNNs. arXiv preprint arXiv:2410.24070.\n\nRedman, W., Bello-Rivas, J., Fonoberova, M., Mohr, R., Kevrekidis, Y. and Mezic, I., 2024. Identifying equivalent training dynamics. Advances in Neural Information Processing Systems, 37, pp.23603-23629.\n\nRibeiro, A.H., Tiels, K., Aguirre, L.A. and Schön, T., 2020, June. Beyond exploding and vanishing gradients: analysing RNN training using attractors and smoothness. In International conference on artificial intelligence and statistics (pp. 2370-2380). PMLR.", "audience_yn": "Yes", "audience_explanation": "This problem is very relevant for the ML field to understand and monitor RNN training.", "requested_changes": "## Main requested changes\n1. The claims about the working of the method should be tested in well-controlled experiments.\nThe claims would be better supported by well-designed numerical experiments based on a fully-understood toy model. For example, to demonstrate the effectiveness of the method, the authors could investigate a simple, low-dimensional system that goes through a bifurcation (e.g. Hopf) that would function as a simplified model of learning in an RNN.\n1. The figure labels should better help the reader to understand the content of the plots and the message overall.\n1. The method should be demonstrated on multiple instances of each RNN parameter combination (architecture, size, etc).\n1. Outside of the captions, more interpretation should be given to the different figures and how the authors arrive at their conclusions/claims from them.\n\n\n\n\n## Smaller remarks\n1. The top and bottom subfigures in Fig. 5b and 5c should be merged, the bottom ones provide no additional information. This is probably also true for Figure 3.\n1. Visualizing high-dimensional data is hard, even in 3D. It might clarify some uncertainties about the full cloud points to use transparency for the figures.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)