Learning Structure from the Ground up---Hierarchical Representation Learning by Chunking
Authors: Shuchen Wu
Noemi Elteto
Ishita Dasgupta
Eric Schulz
Venue: NeurIPS 2022 Accept (2022)
Keywords: Representation Learning
Structure Learning
Cognitive Science
Neuroscience

Abstract: From learning to play the piano to speaking a new language, reusing and recombining previously acquired representations enables us to master complex skills and easily adapt to new environments. Inspired by the Gestalt principle of \textit{grouping by proximity} and theories of chunking in cognitive science, we propose a hierarchical chunking model (HCM). HCM learns representations from non-i.i.d. sequential data from the ground up by first discovering the minimal atomic sequential units as chunks. As learning progresses, a hierarchy of chunk representations is acquired by chunking previously learned representations into more complex representations guided by sequential dependence. We provide learning guarantees on an idealized version of HCM, and demonstrate that HCM learns meaningful and interpretable representations in a human-like fashion. Our model can be extended to learn visual, temporal, and visual-temporal chunks. The interpretability of the learned chunks can be used to assess transfer or interference when the environment changes. Finally, in an fMRI dataset, we demonstrate that HCM learns interpretable chunks of functional coactivation regions and hierarchical modular and sub-modular structures confirmed by the neuroscientific literature. Taken together, our results show how cognitive science in general and theories of chunking in particular can inform novel and more interpretable approaches to representation learning.


Learning Structure from the Ground up-Hierarchical Representation Learning by Chunking
Shuchen Wu
Computational Principles of Intelligence Lab Max Planck Institute for Biological Cybernetics
Tübingen, Germany shuchen. ww@tuebingen.nng mpg.
Noémi Éltetö
Ishita Dasgupta
Department of Computational Neuroscience Max Planck Institute for Biological Cybernetics
Computational Cognitive Science Lab
Department of Psychology
Tübingen, Germany noemi, eltetto@ruebingen.cn
Princeton University dasgupta. www.talgmaillccooooomoooooommame com
Eric Schulz
Computational Principles of Intelligence Lab
Max Planck Institute for Biological Cybernetics
Tübingen, Germany eric. www..cccclleuueeinnn.m.p... de
Abstract
From learning to play the piano to speaking new language, reusing and recombin- ing previously acquired representations enables master complex skills and easily adapt to new environments. Inspired by the Gestalt principle of grouping by proximity and theories of chunking in cognitive science, propose hicrarchical chunking model (HCM). HCM learns representations from non-i.i.d. sequential data from the ground up by first discovering the minimal atomic sequential units chunks. As learning progresses, a hierarchy of chunk representations is acquired by chunking previously learned representations into more complex representations guided by sequential dependence. We provide learning guarantees on an idealized version of HCM, and demonstrate that HCM learns meaningful and interpretable representations in a human-likk fashion. Our model can be extended to Iearn visual, temporal, and iisual-temporal chunks. The interpretability of the learned chunks can be used to assess transfer or interference when the environment changes. Fi- nally, in fMRI dataset, we demonstrate that HCM learns interpretable chunks of functional coactivation regions and hierarchical modular and sub- modular struc- tures supported by the neuroscientific literature. Taken together, our results show how cognitive science in general and theories of chunking in particular can inform novel and more interpretable approaches to representation learning.
Introduction
Sequential data in our everyday life is often hierarchically structured. From streaming this sequential sensory perceptual data, we can identify repeated patterns and bootstrap these to recognize higher order patterns. In cognitive science, identifying repeated, invariant patterns from sequences in units is known as chunking. To get an intuition for chunking, try to read through the following sequence
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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of letters: "DEIKIJKDFFDJJKKDDDD"
Upon reaching the end, if you were asked to repeat the letters from m you might recall fragments of the sequence such as "DF" or "JK". By parsing the sequence of letters only once, you have already detected frequently occurring patterns and memorized them together as units, i.e. chunks. Chunking has been observed a range of sensory and behavioral modalities including language learning [1] [2, action organization [3] and visual perception of structures [5 Chunking as a mechanism is a basis for humans to identify patterns as objects, assigning labels to them to facilitate memory compression [8 91. sequence prediction To M. communication [2] and generalization [4]. Learning hierarchical representations of the world is feature central to human intelligence.
Despite recent success, deep learning models, on the other hand, do not represent explicit hierarchies. Neural networks contain sub-symbolic, nested, non-linear structures whose prediction processes are hard to comprehend. This lack of interpretability raises concerns over their fairness, privacy, robustness and trust- worthiness [15 and manifests itself as key shortcoming of these models To address these shorteomings, researchers have urged to seek inspiration from cognitive science to construct models that resemble the hierarchical and interpretable representations as observed in human learners 12122) We take a two-fold approach to this problem. First, instead of learning from iid data, we ask: what if the time series data that comprises streams of perception comes from a hierarchical structure? Under this assumption, what could be an algorithm that learns the embedded hierarchical structure? We take inspiration from models in cognitive science showing that people perceive structures based on the Gestalt principle of grouping by proximity, and formulate generie hierarchical pattern discovery algorithm that enables the rational discovery of structures with embedded hierarchies. We refer to this model as the hierarchical chunking model (HCM).
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HCM starts out learning a minimal set of units sufficient to explain the sequence and gradually combines these units into increasingly larger and more complex chunks, constructing interpretable hierarchical structures. We derive learning guarantees on an idealized generative model and demon- strate convergence on sequential data coming from this generative model. Thereby, Gestalt principles of grouping can be understood as a rational way of learning representations from sequences with an inherent hierarchical structure. We then show that HCM resembles more to human chunking in qualitative ways compared to recurrent neural network and flexibly transfers components learned from one task to another. We extend HCM to the visual- -emppral domain capable of learning tispal-temporal parts and wholes from higher dimensional sequential data. Taking one step further, we deploy HCM to learn from nigh-dimensional fMRI data, which exerts a hierarchical structure. We demonstrate HCM's interpretable feature extraction ability discover submodules of brain activations directly linkable to behavior supported by the literature.
2 Hierarchical Chunking Model
We define chunk as unit created by concatenating several atomic sequential units together. Taking the training sequence shown in Figure T as an example, the sequence is made up of discrete atomic units from an atomic alphabet set Ag: in this case An {0, 1, A chunk is made up of combination of one or more atomic units in An {0}. 0 denotes an empty observation in the sequence.
Intuitively, if sequence contains inherent hierarchical structure, then there are patterns which span several sequential units sharing these internal structures, examples such sequences are repeated melodies and sub-melodies in music. If the pattern occurs in the sequence, observations between sequential units within the pattern will be correlated. In this case, chunking patterns within a sequence as units simplifies perceptual processing in the sense that the sequence can be perceived one chunk after another, instead of one sequential unit at a time. Furthermore, the acquired "primary" chunks serve building blocks to discover larger chunks that are embedded within the hierarchy of the sequential structure.
Formally, HCM acquires a belief set B of chunks, and uses chunks from the belief set to parse the sequence. HCM assumes that a sequence is generated from samples of independently occurring chunks with probability of (c) evaluated on the belief set B. The probability of observing a sequence of parsed chunks G1, C2, CN can be denoted as P(e1, c2....,CN) = -Io,ez (c.).). Chunks as perceiving units serve as independent factors that disentangle observations in the sequence.

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Figure 1: The Hierarchical Chunking Model. a) Example of hierarchical model generating training sequences. b) Intermediate representation of learned marginal and transition matrices. The most frequent transition that violates the testing criterion is marked in red and can be turned into new chunk. с) HCM combines the two chunks and cR to form a new chunk. d) As HCM observes longer sequences, it gradually learns a hierarchical representation of chunks. HCM arrives the finally chunk hierarchy isomorphic to the generative hierarchy.
The training sequence is parsed by HCM in chunks. At every parsing step, the longest chunk in the belief set consistent with the upcoming sequence is chosen explain the up-coming sequential observations. The end of the previous parse initiates the next parse.
Observing hierarchically structured sequence as illustrated in Figure Th HCM gradually builds up a hierarchy of chunks starting from an empty belief set. It first identifies set of atomic chunks to construct its initial belief set IB. Initially, these will be chunks of length one, yielding one-by-one processing of the primitive elements.
For one belief set B. HCM keeps track of the marginal parsing frequency M (c;) for each chunk c; in B, a vector with size and the transition frequency between chunk c; followed by chunk cjs as illustrated in Figure Tp. Entries in M and T are used to test the hypothesis that consecutive chunk parses have correlated consecutive occurrence within the sequence via X² indeepeedeee test. If two chunks and have significant adjacency dependence based on their entries in M and T. they are chunked together to become e₁ @ en. which augments the belief set by one. One example of chunk merging is shown in Figure T
Independence Test We use 2test of independence to assess the correlation of consecutive occurrences of ez followed by CR. Let ez be indicator variable that is when chunk q is parsed and 0 otherwise, similarly we formulate as another indicator variable of parsing the chunk c,. We evaluate the 2a-ule criterion reject the null hypothesis that the consecutive observation of q followed by c,- is statistically independent:
N(p(e,, CR) - p(cz)p(cR))) p(Cr.)((CR)
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p(cr., CR) and p(cr.)) (PCR) are evaluated from M an T. The degree of freedom is 1. A x² -probability of less than 0.05 is the criterion reject the null hypothesis (i.e. that a and c, occur independently).
There are two versions of HCM. The Rational Chunk Learning HCM learns chunks in an idealized way which to study learning guarantees. The online version of HCM is approximation

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Figure 2: a) Example graph generated from the hierarchical generative model with a depth of = 5. b) Learning performance of HCM and RNN with increasing training length and for increasing depths. Performance was averaged over 30 randomly-generate graphs.
to the rational HCM that can be adapted to different environments and processes sequences online. Pseudo-code for both algorithms can be found in the SI.
Rational Chunk Learning: HCM Ideal Observer HCM is initiated with empty belief set and it first finds a minimally complete belief set after the first sequence parse. In each iteration, the entire sequence is parsed to evaluate M and T, which used to find consecutive chunk parses in the existing belief set that violate the independence testing criterion. From these dependent chunk pairs, the pair with the largest estimated joint probability is combined into new chunk. The new chunk enlarges the belief set by one. The chunks in the new belief set are used to parse the sequence in the next iteration. This process repeats until all of the chunks in the belief set pass independence halting criterion, which measures if all of the chunks in the belief set are currently independent, again assessed via x 2test (see SI).
Online Chunk Learning The online chunk learning HCM approximates the ideal observer HCM by learning new chunks when the training sequence processed on the go. To have a feature that encourages adaptation to new environmental statistics, entries in M and T can be subject to memory decay. We use the ideal observer model to demonstrate learning guarantees, but use the online model to learn representations in realistic and more complex set-ups.
2.1 HCM Learns Representations from the Ground Up
As HCM learns from sequence, it starts with no representation and gradually builds up interpretable representations described by chunk hierarchy graph 9 with the vertex set being the chunks and edges pointing from constituents to composites. Shown in Figure T is the gradual build-up of one such graph as the model learns from a training sequence coming from the generative hierarchy in Figure Ta. At = 10, HCM learns only the atomic chunks, at = 60, HCM has already constructed two additional chunks; when & = 100, two more additional chunks are constructed. HCM arrives at the final chunk hierarchy t 150.
3 Generating Sequences with a Hierarchical Structure
We construct generative model to study HCM's behavior formally and empirically. The generative model constructs random chunk hierarchies from which non-iid sequences are sampled. Such graph Ga contains vertex set VA, and edge set Es, to describe the relation between chunks and their constituents. One example is illustrated in Figure Th. Ag is the set of chunks used to construct the sequence. The depth d specifies the number of chunks created in the generative process.
Starting with an initial set atomic chunks Ag. at the i-th iteration, two chunks cL, CR are randomly chosen from the current set of chunks A; and are concatenated into a new chunk e, 3 cg. augmenting A; by one to A++1- Meanwhile, an independent occurrence probability is assigned to each chunk under the constraint that the probability of occurrence for every new chunk c; in the construction process evaluated on the support set A carries the largest probability mass.
Once graph hierarchy is constructed, we construct non-iid observational sequences by consecutively sampling chunks from the hierarchy with their corresponding probability, under the constraint that no two chunks with child chunk are sampled consecutively.

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3.1 Learning Guarantee
Theorem:
As the length of the sequence approaches infinity, HCM learns hierarchical chunking graph isomorphic to the generative hierarchical graph g.
Proof Sketch: We approach this proof by induction. Further details can be found in the SI. Base step: The first step of the rational chunking algorithm is find the minimally complete atomic set chunks form initial belief set. This procedure guarantees that Go Go- Additionally, the probability mass of the learning model at step O and the generative model at step 0 is asymptotically the same as the sequence length approaches infinity. Induction hypothesis: Assume that the learned belief set B3 at step i contains the same chunks as the alphabet set A; in the generative model, the chunk combination pair with the biggest evaluated joint occurrence probability violating the independence test is picked to be concatenated into chunk to extend the belief set: this chunk is the same chunk node created by the hierarchical generative model. End step: The chunk learning process stops once the independence criterion is no longer violated. This is the case once the chunk learning algorithm has learned belief set B Add
3.2 Learning Convergence and Comparative Data-efliciency
To evaluate and show HCM's learning performance, we trained HCM to learn hierarchies of chunks from sequences generated by the hierarchical generative model. Shown in Figure HCM's learning performance sequence length increases, averaged over 30 independently generated random graphs with the same depth d. One example of such graphs shown in Figure 2.. Kullback-Leible divergence used evaluate learning performance. To this end, learned hierarchies by HCM were used to generate sequences, which were then evaluated on the support set of the alphabet set in the generative model.
Figure 2 shows the KL-divergence between the learned and ground-truth distribution for increasing depths of the generative graphs. For each depth, the KL-divergence was evaluated on 30 random generative models with sequence length increasing from 50 to 3000. Overall, the KL-divergence decreased the training sequence length increased and converged with longer training sequences, showing a closer representation resemblance to the generative model.
A similar training and learning evaluation was conducted on 3-layer Recurrent Neural Network (RNN) with hidden units for comparison. the length of the training sequence increased, the KI. .ivveieeeee of RNN converged at slower rate than HCM. This competitive advantage in data efficiency became more pronounced with increasing depth of the generative hierarchy.
HCM Resembles Human Chunk Learning
Here we compare the chunk learning behavior HCM to the learning characteristics of humans. To that end, we used data collected from sequence learning study by [23] with 47 participants under the license CC-BY 4.0. As shown in Figure 3h, the training sequence comprised chunks ABC and D, independently occurring with equal probability. The study assessed how participants built up chunk knowledge gradually. Participants' reaction times reflected that, after enough training, they were anticipating several upcoming sequence elements, suggesting that they have acquired longer chunks
Figuue 3p. left) (3
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In a similar vein, we measured online chunk size increase of HCM using the same method as in [23] and, for comparison, RNNs (see SI for further comparisons other algorithms). HCM. similarly to humans, started learning longer chunks early in the sequence. By contrast, RNN did not start to chunk until after step 300, and when started to learn chunks, the increase rate of the predictive horizon was not as steep as HCM's. Evaluating the average rate of chunk growth also showed that HCM builds up chunks as learning progresses was more similar to participants' than the RNN's (Figure 3b). The negative log-pnn ong-robbbbilitis of sequence elements generated by the HCM and RNN were both significantly related to human reaction times (that reflect the certainty of their internal predictions (24). Yet, the relationship was substantially stronger (Figure 3)1) between HCM (3 = 16.74, p S 0.001, T = 0.165, BIC = 313586.5) and human participants compared to that 0.085, BIC 31 314236.4). These results suggest that HCM resembles human chunk learning more strongly than RNNs and can therefore be seen as the cognitively more plausible approach hierarchical representation learning.
of the RNN (3 9.24, p < 0.001,
HCM Permits Transfer Between Environments
One characteristic of human learning is that previous learning experience facilitates and sometimes interferes with acquiring a new skill [25). Having an interpretable representation can inform us about positive or negative transfer priori.
An HCM has learned a chunking graph in Figure 4 from an enviromment. When it switches to another enviromment with generative model overlapping with its previously acquired representation, can reuse the learned subgraphs of chunks marked in gray as in Figure 4p and learns faster than a naive HCM in Figure 4c. Vice versa, transfer can be detrimental when there is between the learned chunking graph and the generative model of the new environment. For the same HCM as in Figure 4h, transferring to environment with chunking graph in Figure 4t implies learning the shaded chunks anew, in addition to running the risk of being misled by the previous representations. As result, the early performance of the pre-trained HCM suffers more from an interfering environment than a naive model in Fe. Interpretability of HCM's representations enables the assessment of facilitation interference when the environment changes.
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Figure 4: a) Example of a representation leamned by an HCM. b) A facilitative environment with a generative model overlapping in its structure with the test environment. Gray shadows mark the chunks that can be directly transferred. c) Average performance over the first 500 trials after the environment switches. d) Interfering environment. Gray shadows marks chunks that need to be acquired anew. e) Average performance over the first 500 trials after the environment switches.
6 Generalizing to Visual Temporal Chunks via the Principle of Proximal Grouping
Humans excel at finding structures in hicrarchical visual objects and grouped movements. The Gestalt principle of grouping by proximity suggests that we tend to group objects that close to one another into a cohesive unit 26/22] This principle has been suggested to play key role in human perceptual grouping 228), benefits working memory (29] and reduces visual complexity (3). Indeed, in humans and other animals, learning of adjacent relationships prevails over non non-adjacent ones [31]. Therefore, the adjacent dependency structure be expanded to chunking in visual temporal domains [32]. To emulate this ability of chunking via proximal grouping, we extend HCM to learn visual temporal chunks.
Visual temporal chunks subsume temporal length and varying visual slices in each temporal slice Figure 5l)). One imagine visual temporal chunk as having 3D shape the first two

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a) Visunl-Tempural Chunking
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Figure 5: a) HCM learns visual-temporal chunks by extending the transition matrix to take account of time differences. b) Left: A visual hierarchical model where complex images are composed of simpler images. Right: Initial, intermediate, and complex chunks learned by HCM trained on sequences of images sampled from the visual hierarchical model.c) Top: A GIF of moving visual used as a sequence to train HCM. Bottom: Examples of temporal and visual chunks learned by HCM.
dimensions are the visual part of the chunk, and the object's length is the temporal part, made of stacked tesporltemporal pixels. Within each temporal slice are the visual features identified by the chunk. As the model iterates through data across its temporal slice, the chunk that attains the biggest visual temporal volume explains part of the observational sequence. Multiple visual temporal chunks can occur simultaneously. Starting at the visual temporal time point marked by the previous chunk, chunks are identified and stored in M. The transition matrix modified to account for the temporal lag difference between adjacent chunk pairs within proximity parameter and records the frequency that one chunk transitions into another for each time lag. Whenever a pair adjacent chunks are identified, an independence hypothesis test evaluates whether the adjacent observation correlated. Chunks that violate the hypothesis test are combined to parse future sequences.
Learning Part-Whole Relationship Between Visual Components We show HCM learned chunks in the visual domain from sequence of independently sampled images. Figure 5p left shows hierarchical generative model the pixel-wist image domain. set of elementary visual units in the lowest hierarchy level combines into intermediate and more complex visual units higher up in the hierarchy. All of the constructed elements in the hierarchy occurred independently according to probability drawn from Dirichlet flat distribution. Images in the hierarchy were independently sampled from the generative distribution to become the training sequence. In Figure 5p, right we show the chunk representations learned by HCM different stages. Initially, HCM acquires the individual pixels as chunks to explain the observations. As HCM proceeds with learning, it discovers visual correlations among the pixels and constructs increasingly complex visual parts.
Learning Visual-Tempore Movement Hierarchies Instead of seeing image after another sampled from an independent, identically distributed distribution, real-world experiences contain correlations in both the visual and temporal dimensions. From observing object movements across space and time, the visual system learns structures from correlated visual and temporal observations, decomposes motion structure, and groups moving objects together as whole [33 To emulate this
type of environment, animated GIF of squid swimming in the sea (Figure 5e) was used as -isual-temporal sequence train HCM. As learning advances, HCM learns chunks spanning both the visual and temporal domains. There are visual-temporal chunks that mark the movements of tentacle and the rising-up motion of bubble. Additionally, visual chunks resemble a part of visual's eye and face. The meaningful chunks in the visual-temporal domain suggest the grouping principle enables the plausible learning movement sequences and aids the perception of objects as wholes and their corresponding parts.
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Figure Application avenues of HCM on fMRI data. a) Example chunks of brain functional activation regions. b) HCM learns hierarchical functional network with bigger chunks emerging from its constituents. e) Chunk activation patterns responding to scene content Distinct response of retrieved chunks to tagged scenes. e) Average chunk size across age groups.
Learning Hierarchies of Brain Activation from Resting-state fMRI data
HCM learns hierarchies from structured sequential data. As brain activation has been suggested to be hierarchically structured (34), we demonstrate HCM's usefulness to learn structures in biological neural networks activating in response complex stimuli by running HCM on resting-state -tate fMRT data set.
We used developmental data set provided by the nilearn package with BSD License 33 and originally collected by (36] with its corresponding IRB approval. This data set contains the resting- state BOLD activity of 155 participants ranging from age 3 to 40, while watching the silent movie "Partly Cloudy". BOLD signal was extracted from functional brain regions defined by the MSDL Atlas (37), with confounds excluded and transformed into rounded, normalized time series.
HCM's Chunks Reflect Structural, Functional and Anatomical Connectivity Figure 6 shows three typical examples of learned chunks for randomly- participant. The labels of functional regions come from the MSDL atlas [37]. The first example is the co-activation of D ACC and R A Ins. co-activate in the presence of emotions, pain, and humor.
These two regions have been observed They have been suggested to be a key hub of the salience network [1. The second example chunk contains the activation of R DL.PC and Front Pole. These regions belong to the visual attention network and are known to be anatomically connected [42[44] A final example is the chunk of L Ins and Cing, which are also known to be anatomically and functionally connected [45). Thus, the chunks discovered by HCM correspond to empirically-vvrrifed patterns of functional activity.
HCM's Chunks Recover Hierarchical Activation Patterns In fMRI data, hierarchies of chunk activation constructed by HCM reflect networks of functional regions. On the top of the hicrarchy, the largest chunk contained L DLPFC, D ACC, R A Ins, Cing, and L Ant IPS (Figure 6D). Those regions are known to co-activate during cognitive tasks that demand attention, working memory, and control [1 Chunks in the intermediate levels of the hierarchy reflect sub-networks of functional connectivity. Sub-chunks such as ACC, Ins, L Ant IPS, and DLPFC have been suggested to conjointly activate in cognitive effort--elated activities 41 Atomic chunks in the hierarchy such as D ACC and L Ant IPS activate individually sometimes without their parent chunks. Indeed, they have distinct functional signatures for affect processing [66 47 and visual attention control [48] Upon exposure a time series in fMRT data, HCM constructs chunks from their constituents and arrives at a hierarchy of chunk relations, indicating nested network structures in the human brain.
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HCM's Chunks can be Matched with Stimulus Onsets The retrieved chunks by HCM can be tagged with critical stimulus onsets. We tagged 19 critical moments involving social and emotional content in the movie. Figurel6p shows one example chunk activation upon stimulus onsets. Frontal DMN, right frontal pole, ventral anterior cingulate cortex, and right anterior insula activate together as recurring unit after participants witness a scene with characters greeting and hugging each other.
These regions have been suggested to be involved in social and cognitive processing [49]. Another example is the activation of areas known to be involved in emotion and language processing: D ACC and Broca [50], during a scene containing social interactions. In the meantime, the left and right prefrontal cortex, involved in theory- of-iidd [36,, also lights up.
We categorized the tagged moments into 3 groups of different emotional load: sadness, anger, and compassion. We then looked at the activation probability of retrieved chunks within the 6 seconds after watching those tagged scenes. Figure [6 shows list of such chunks from one participant with their activation probability for each emotional category. For example, the left and right insula, known to be involved in affective processing [51], have a 0.4 activation probability after witnessing scenes of sadness or anger, but no activation after witnessing scenes of compassion. The same holds for R
DLPFC and R Par that have been documented to activate in response to emotional conflict [52]. On the other hand, regions such as R TPJ and R pars opercularis that are involved in emotional reactions and theory of mind processes [53], activate in response to a scene of compassion or anger, but not to scene of sadness. Thus, chunks of active brain regions can be related to complex stimuli, and regions activate selectively in response to one or more categories of emotional stimuli, but not others.
HCM's Average Chunk size Correlates with Participants' Age HCM can also be used to perform meaningful analyses at the population level. Specifically, we find that HCM's returned average chunk size per participant correlates significantly with age (Figure 6ş). The older the participants are, the longer are the chunks found in their data (r = 0.23, p S 0.001). This discovery is in line with findings in the original study, which showed increase in modularization of ToM and pain circuits across development (36].
To summarize, we applied HCM to learn chunks from developmental fMRI data set. HCM enabled the discovery of spatially and temporally correlated activation chunks that are theoretically and empirically meaningful. The resulting chunks can be linked to complex stimuli and offer directly interpretable insights into the structure and function of brain activity.
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Related Work
HCM extends upon decades of previous cognitive science and psychology research on chunking. In cognitive science, process models such as PARSER and competitive chunking were demonstrated to generate qualitatively similar chunks as in human sequence learning (44 555) HCM a rational algorithm that learns the underlying chunks when the sequence is generated from a hierarchical chunking graph. Therefore, the chunking criterion is no longer a heuristic but rational learning strat- egy that enables hierarchical structural discovery. On top of inheriting the merits of its predecessors, HCM generalizes the chunk learning principle to higher dimensional sequential domains such as visual-temporal sequential data.
HCM relates to several other lines of research. One is program induction. In program induction, explicit representations are acquired by searching for programmatic structures that best explain observational samples (22), and consolidating these offline with library learning [56). However, domain expert knowledge is needed to specify the primitive programs; the relations and composition rules must adapt to the task settings and sensitively influences the quality of retrieved representations. Other approaches to structure learning include unsupervised parsing (57]. which learns a stochastic and-or graph from sequential data. HCM is distinct in adapting its representation granularity to discover bigger chunks from data without pre-specifying the structure.
Another category of models to learn from sequential data are traditional sequence learning models including Hidden Markov Models (HMM), n- n-gram models and their variants to capture multi-scale sequential structure such as Hidden semi-Markov model (58) and hierarchical HMM 59. The parameters of these models proliferate exponentially as function of chunk length, implying memory inefficiency. Additionally, these models demand a structure specification before fitting parameters to the data. They also lack the adaptive recombination and reuse of pre-existing components. The

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same issue is with neural network approaches to extract chunks from sequences 600 62]. Apart from lacking in interpretability. these models do not leverage the concatenation process observed in humans or reusing previously learned representations to construct new representations.
The principle of iterative merging of chunks has been used in compression algorithms, such tok- enizing methods in NLP, which were developed to optimize sequential data compression. Tokenizing methods such as Byte-Pair Encoding [63] iteratively merges the most frequent pairs of chunks to build a vocabulary of a text corpus. This objective is easy to compute but gives rise to ambiguous parses of the text (e.g. [AB, C)/ BCJ) To minimize parse ambiguity, WordPiece [64] merges chunks that increase the likelihood of the corpus the most. However, the objective of WordPiece is expensive to compute. HCM circumvents the problem of computing the global sequence likelihood by instead maximizing the local chunk continuation likelihood. The computational efficiency of HCM makes a plausible cognitive model of chunking as well as a promising method for NI.P.
Probabilistic context-free grammars (PCFGs) are related to HCM in that they use trees as a represen- tational form of sequences. The parse trees of PCFGs denote production rules, such as S + NP + VP. These production rules define how abstract syntactic units (non-terminals), such as a noun phrase and verb phrase, are instantiated into concrete string of words (terminals) to compose a sentence, such as "we wrote the paper'. In comparison, the generative tree of HCM denotes statistical relationships among concrete chunks, such as "wrree' ''the paper'. Extending HCM to represent abstraction is an exciting future direction, on which avenue the comparison to PCFGs will be instructive.
Discussion
Our work has its limitations. Currently, we fix the memory decay and the deletion threshold parame- ters to a priori plausible values. In future work, these parameters could be adapted online based on environment volatility. Another limitation is its scalability: at the moment, HCM learns represen- tations from semi-high dimensional sequential data (i.c., currently between 1 to 625 dimensions). We are actively looking into generalizing this algorithm higher dimensional data domains by combining it with existing neural network approaches or computer vision algorithms such coherent point drift (65] or normalized cuts (66] to allow for the learning of ambiguous and high dimensional chunk exemplars. It is also possible to combine HCM with the compressed representation, such as the hidden activity of an auto-encoder to process and learn the structure from downstream representation. In this work, HCM learns one type of hierarchy of compound representations. However, we can show that HCM can be generalized to not only learn simple chunks but also chunks in projected spaces and thereby generalize between two chunks that contain the same motif (for example, "1222121 12"" and "34443434"; see SI for detailed results). In the future, it might be worthwhile to further combine our approach with others amongst taxonomy of representational hicrarchies.
HCM also opens up other application directions. One direction integrating HCM with deep neural network approaches as an interface between human understanding and distributed computation. Learning hierarchies of coherent activations from intermediate hidden units has the potential to reveal neural networks' underlying computation structure. Furthermore, it is also possible to equip HCM with additional top-down encoded representations, for example, by pre-training on other sequences or by adjusting the chunks by hand before the training starts. Another direction in neuroscience or behavioral research is to learn chunks of tagged animal movements that enable insights into the emergence of behavioral structure (67). Finally, finding patterns that form cognitive unit is vital task for infants to learn about the structures of the world and resembles the process of formulating scientific theory from observation (68). HCM can function as means to come up with world models by observation, ready for experimental interventions or active learning to delineate the causal structure within 669.
10 Conclusion
We have proposed a hierarchical chunking model (HCM) that learns chunks from non-iid sequential data with a hierarchical structure. HCM starts out learning an atomic set of chunks to explain the sequence and gradually combines them into increasingly larger and more complex chunks. The output of the model is dynamical graph that is a trace of the evolving representation. The resulting representations are easy to interpret, and flexibly reusable.
10

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16


Reviews:
[{"review_id": "thYSjgbHBSJ", "reviewer": "Reviewer_TVcV", "summary": "This paper presents a hierarchical chunking model (HCM), which learns chunks from non-i.i.d. sequential data. The full learning process elicits a trace of building hierarchy from the ground up, similar to humans' perception and mind process, and hence human-interpretable. The proposed model is effective, adaptive, and successful in considered applications spanning from symbolic sequences to visual temporal data, from artificial settings to real-world problems like learning hierarchies of brain activation from resting-state fMRI data.", "questions": "1. Is it possible to have ambiguity in the process of parsing larger and larger chunks, e.g., what if there are multiple largest probabilities in the transition matrix (not necessarily an exact tie, but close)? Would one choice of breaking the (approximate) tie greatly change the later process? The whole procedure seems conceptually similar to hierarchical clustering, which merges small parts in a greedy fashion. Then, is the final result going to be sensitive to some \"noisy\" decisions made earlier? Here, a \"noisy\" decision may be a choice of breaking the tie as mentioned above. But, in addition to that, can the authors also explain if the final result tends to be sensitive to a small phase shift (e.g., start reading from the second item instead of the very beginning of a sequence)? Is it sensitive to noisy observations?\n\n2. In Figure 2a, it seems 0 is treated differently from other symbols, why so and is this kind of treatment always needed for empty observation?\n\n3. This paper presents several cases where the model is capable of discovering known patterns. While these are necessary to show the model's efficacy when tested against ground truth, has the model also exhibited its ability to discovery new patterns, e.g., in fMRI data?", "limitations": "Limitations and their connections to related and future work were adequately addressed.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "This paper is well written, whose ideas are easy to follow. The goal, motivation, and scope is clear; theory and algorithm parts are solid; experiments are well designed and conducted, spanning a diverse range of topics; experimental results are significant, which are further interpreted and discussed thoroughly. There are two things I would suggest the authors considering improving.\n\nHow the work presented in this paper is positioned among other related work in the literature is not clear to me until the final discussion section. However, comparisons were only discussed in general terms, largely focusing on the methodological perspective. If any of the techniques introduced in this paper are derived, influenced, or inspired by prior work in the similar field, could the authors include comparisons regarding performances on the considered experiments in the paper? The only experimental comparison in this paper seems to be with RNN (and human performance), which is methodologically quite different from the proposed HCM. In fact, I think comparing to RNN and showing HCM's superiority (e.g., as in Figures 2 and 3) is relatively weak, as HCM is especially built for the purpose of studying those internal sequential structures. For instance, the similar way how HCM and generative model in Sec 3 are built, should naturally make HCM work well for that type of generative model. So, to me, experimentally (rather than algorithmically) comparing HCM to more related other algorithms (like the ones mentioned in the discussion section) will be more convincing.\n\nFigures are generally well designed, however, their layouts and arrangements are not that clean to me. For example, in Figure 1, it took me a while to figure out where the different subfigures are, and it seems to me there are overlaps between subfigures.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "mEgWI4tpsG1", "reviewer": "Reviewer_SQe7", "summary": "The authors proposed and tested a Hierarchical Chunking Model. This model discovers a hidden hierarchical structure while having a stream of state transitions as inputs. By processing a linear stream of one-dimensional information as transitions between sets of different lengths of sequences, the model starts to build a hierarchical structure. Human participants also use this chucking strategy. The hierarchical structure inferred from changes in the reaction times is closely related to the model-generated structure. Last, the authors provide various examples in which sequential information processing can get benefit from this human-like chunking strategy.", "questions": "Is there any threshold determining which level of correlation is good enough to have a new chunk? And how the memory decay parameter and the deletion threshold are determined? I may miss how they were determined. Are they predetermined as fixed values? Or, these can also be determined by another hierarchical variable which tracks to what extent the environment is volatile. In this way, the model may be able to reduce the learning time.\n\n \n", "limitations": "The current model may not be able to generalize the structure learned from a sequence if the stimuli changes while keeping the same structure. (e.g., 12221212 -> 34443434).", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths. By constructing a hierarchical structure, the model can efficiently learn a novel sequence if it was generated from the same structure and afford generalization. This hierarchical chunking model doesn't require a prior model structure but can be generated by the data itself. Interestingly, without knowing the outer stimuli causing these activations, this model could capture the meaningful neural networks associated with different cognitive states.\n\nPotential weaknesses. The model only learns the sequence determined by the deterministic transition. This may capture better some types of sequences (where a temporal order matters). However, this can be more flexible if the model can learn the hidden structure generating a chunk of sequence. For example, if a sequence is generated by a random walk, the current model would increase the number of independent chunks, while it could be better to be captured by a certain network structure describing the states of each of the communities and how they are connected. \n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "X8jtYn9oPNf", "reviewer": "Reviewer_2nnD", "summary": "The authors give a statistical algorithm for hierarchical chunking of data tokens.  They prove that when their algorithm's structure matches an underlying data-generating process, it can and will learn the correct structure.  They then demonstrate the interpretability of the learned chunkings, with application to cognitive science.  They perform simple experiments with visual, temporal, and visuotemporal chunking, and demonstrate that an application of their algorithm to fMRI data finds chunks corresponding to known areas of the brain.", "questions": "When the authors write about the Gestalt principle of grouping by proximity, is this meant to reflect proximity in topographic neural maps, egocentric space, or allocentric space?\n\nWhat is the big difference between learning a semi-Markov HMM (Hidden Markov model) or hierarchical HMM with a likelihood exploiting the Size Principle (https://www.sciencedirect.com/science/article/abs/pii/S0001691809001553) and the actual HCM?\n\nDo the independence tests used in the HCM have a relation to an optimization criterion, such as total correlation or similar?  Why use p<0.05, since we generally want stronger guarantees in an automated system?\n\nWhat is the beta-test statistic mentioned on line 178-179?\n\nWhat did the authors mean by an \"overlapping representation\" in terms of switching environments at line 187?  Representations are typically a part of some internal model, not a part of the environment itself.", "limitations": "The authors have mostly addressed my questions and concerns, but nonetheless, using a false-positive rate of 5% when you expect to form more than 20 chunks means you expect there to be at least one false-positive chunk.", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n* Originality and clarity of the statistical justification for chunking\n* Strong experiment comparing the HCM to an RNN for predictive convergence\n\nWeakness:\n* Incorrect use of p-values\n* Too much of how the HCM actually works is in the appendix: pseudo-code, halting criteria, etc.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "OEZ0vOwsiYp", "reviewer": "Reviewer_ybpV", "summary": "This work proposes a model that learns hierarchical chunks from non-i.i.d data in an unsupervised fashion. The central part of the model uses a statistical test to assess the dependence between consecutive items in a sequence. The authors use a sequence learning task to show that the model learns more similarly to humans than RNNs, and predicts human reaction times more strongly. This leads them to conclude that the HCM is more cognitively plausible than an RNN. They also show the performance of the model on visual sequences and resting state fMRI data, and illustrate the ability of the model to do online learning.", "questions": "* What are the memory demands of the model? A formal analysis would be useful, given the vague claim that the HCM can handle \"semi-high dimensional\" data. This is also of interest because one of the uses for chunking may be in memory compression.\n* How did you choose the size and architecture of the RNN for comparison with the HCM?\n* The text suggests that the online HCM can be subject to memory decay to adapt to changing statistics. Is a significant decay / deletion threshold actually required for any of the present experiments?\n* Supplemental Section G gives a hyperparameter for the HCM reaction time analysis (the probability of within chunk RT is 1 - 4/eps). Why was this value chosen? What happens when other values are chosen?\n* What are the implications of actual value of the statistical threshold? Is it possible that different contexts may lead learners to have more strict or relaxed thresholds?\n\nSuggestions (from major to minor)\n* Dedicate a separate section to Related Work.\n* Figure 1 does not define the variable d as depth.\n* In the Supplemental Material E.1.3, Equations 23 & 24 appear to be identical.", "limitations": "Edit: the authors have addressed these items on the checklist.\n\nThe authors did not complete items 5a-5c on the checklist. I will rate this paper a reject as the ethical concerns are not completely addressed.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Edit: I thank the authors for their extensive replies to the reviews. They have addressed several concerns, particularly with their proposed  changes to summarize related work in reply to both me and reviewer TVcV. For now I have increased my score to a Weak Accept (6). This may change with any continued discussion.\n\nThe experiments are varied and illustrate different possible uses of the model. In particular, the extension of the HCM to spatiotemporal data and to online learning are both intriguing areas of future research. I also appreciate the pursuit of cognitively plausible models, the proof of recoverability, and the discussion of the importance of hierarchical representation learning in human intelligence.  The paper and method are also explained clearly, though some details about experimental choices are missing (even when referring to the supplement).\n\nI do find the discussion of how this model relates to other previous approaches quite lacking. This makes the originality and significance of the proposal less clear, as some related work is missing. They write that \"other types of hierarchical representations such as clustering or relational hierarchies are out of the scope for this work.\" Some clustering algorithms are certainly related and within scope. In NLP models, tokenizer methods such as Byte-Pair Encoding or WordPiece can be thought of as clustering approaches that, like chunking, facilitate compression of sequential data. Furthermore, it would be useful for the authors to discuss the relationship between their definition of a hierarchical structured sequence and a probabilistic context free grammar. They look quite similar to me, and much theoretical work has been done to study the ability of various models to capture different types of artificial languages (e.g. Hewitt et al. 2020, RNNs can generate bounded hierarchical languages with optimal memory). This discussion may even facilitate the authors' comparison of the HCM to other models.\n\nGiven this and some of the lacking details, I am rating the contribution of this paper as fair. If the ethical concerns are addressed, I would rate this as a 5 (borderline accept). I do think the framing of this work to connect it to the cognitive science literature on chunking is useful, and I am willing to increase this score if my concerns and questions are addressed.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept