Original: Learning Structure from the Ground up—Hierarchical Representation Learning by Chunking
Revised: Learning Structure from the Ground up—Hierarchical Representation Learning by Chunking

Original: Abstract
Revised: Abstract

Original: From learning to play the piano to speaking a new language, reusing and recombining previously acquired representations enables us to master complex skills andeasily adapt to new environments.
Revised: From learning to play the piano to speaking a new language, reusing and recombining previously acquired representations enables us to master complex skills andeasily adapt to new environments.

Original: Inspired by the Gestalt principle of grouping byproximity and theories of chunking in cognitive science, we propose a hierarchicalchunking model (HCM).
Revised: Inspired by the Gestalt principle of grouping byproximity and theories of chunking in cognitive science, we propose a hierarchicalchunking model (HCM).

Original: HCM learns representations from non-i.i.d.
Revised: HCM learns representations from non-i.i.d.

Original: sequentialdata from the ground up by first discovering the minimal atomic sequential units aschunks.
Revised: sequentialdata from the ground up by first discovering the minimal atomic sequential units aschunks.

Original: As learning progresses, a hierarchy of chunk representations is acquiredby chunking previously learned representations into more complex representationsguided by sequential dependence.
Revised: As learning progresses, a hierarchy of chunk representations is acquiredby chunking previously learned representations into more complex representationsguided by sequential dependence.

Original: We provide learning guarantees on an idealizedversion of HCM, and demonstrate that HCM learns meaningful and interpretablerepresentations in a human-like fashion.
Revised: We provide learning guarantees on an idealizedversion of HCM, and demonstrate that HCM learns meaningful and interpretablerepresentations in a human-like fashion.

Original: Our model can be extended to learn visual,temporal, and visual-temporal chunks.
Revised: Our model can be extended to learn visual,temporal, and visual-temporal chunks.

Original: The interpretability of the learned chunkscan be used to assess transfer or interference when the environment changes.
Revised: The interpretability of the learned chunkscan be used to assess transfer or interference when the environment changes.

Original: Finally, in an fMRI dataset, we demonstrate that HCM learns interpretable chunks offunctional coactivation regions and hierarchical modular and sub-modular structures confirmed by the neuroscientific literature.
Revised: Finally, in an fMRI dataset, we demonstrate that HCM learns interpretable chunks offunctional coactivation regions and hierarchical modular and sub-modular structures confirmed by the neuroscientific literature.

Original: Taken together, our results showhow cognitive science in general and theories of chunking in particular can informnovel and more interpretable approaches to representation learning.
Revised: Taken together, our results showhow cognitive science in general and theories of chunking in particular can informnovel and more interpretable approaches to representation learning.

Original:  Introduction
Revised:  Introduction

Original: Sequential data in our everyday life is often hierarchically structured.
Revised: Sequential data in our everyday life is often hierarchically structured.

Original: From streaming this sequentialsensory perceptual data, we can identify repeated patterns – and bootstrap these to recognize higherorder patterns.
Revised: From streaming this sequentialsensory perceptual data, we can identify repeated patterns – and bootstrap these to recognize higherorder patterns.

Original: In cognitive science, identifying repeated, invariant patterns from sequences in unitsis known as chunking .
Revised: In cognitive science, identifying repeated, invariant patterns from sequences in unitsis known as chunking .

Original: To get an intuition for chunking, try to read through the following sequenceof letters: “DFJKJKJKDFDFJKJKDFDF”.
Revised: To get an intuition for chunking, try to read through the following sequenceof letters: “DFJKJKJKDFDFJKJKDFDF”.

Original: Upon reaching the end, if you were asked to repeatthe letters from memory, you might recall fragments of the sequence such as “DF” or “JK”.
Revised: Upon reaching the end, if you were asked to repeatthe letters from memory, you might recall fragments of the sequence such as “DF” or “JK”.

Original: Byparsing the sequence of letters only once, you have already detected frequently occurring patternsand memorized them together as units, i.e. chunks .
Revised: Byparsing the sequence of letters only once, you have already detected frequently occurring patternsand memorized them together as units, i.e. chunks .

Original: Chunking has been observed in a range of sensoryand behavioral modalities including language learning [1, 2], action organization [3, 4] and visualperception of structures [5–7].
Revised: Chunking has been observed in a range of sensoryand behavioral modalities including language learning [1, 2], action organization [3, 4] and visualperception of structures [5–7].

Original: Chunking as a mechanism is a basis for humans to identify patternsas objects, assigning labels to them to facilitate memory compression [8, 9], sequence prediction[10, 11], communication [12, 13], and generalization[14].
Revised: Chunking as a mechanism is a basis for humans to identify patternsas objects, assigning labels to them to facilitate memory compression [8, 9], sequence prediction[10, 11], communication [12, 13], and generalization[14].

Original: Learning hierarchical representations ofthe world is a feature central to human intelligence.
Revised: Learning hierarchical representations ofthe world is a feature central to human intelligence.

Original: Despite recent success, deep learning models, on the other hand, do not represent explicit hierarchies.
Revised: Despite recent success, deep learning models, on the other hand, do not represent explicit hierarchies.

Original: Neural networks contain sub-symbolic, nested, non-linear structures whose prediction processesare hard to comprehend.
Revised: Neural networks contain sub-symbolic, nested, non-linear structures whose prediction processesare hard to comprehend.

Original: This lack of interpretability raises concerns over their fairness, privacy, robustness and trust-worthiness [15–18] and manifests itself as a key shortcoming of these models
Revised: This lack of interpretability raises concerns over their fairness, privacy, robustness and trust-worthiness [15–18] and manifests itself as a key shortcoming of these models

Original: To address these shortcomings, researchers have urged to seek inspiration from cognitivescience to construct models that resemble the hierarchical and interpretable representations asobserved in human learners [21, 22].
Revised: To address these shortcomings, researchers have urged to seek inspiration from cognitivescience to construct models that resemble the hierarchical and interpretable representations asobserved in human learners [21, 22].

Original: We take a two-fold approach to this problem.
Revised: We take a two-fold approach to this problem.

Original: First, insteadof learning from iid data, we ask: what if the time series data that comprises streams of perceptioncomes from a hierarchical structure?
Revised: First, insteadof learning from iid data, we ask: what if the time series data that comprises streams of perceptioncomes from a hierarchical structure?

Original: Under this assumption, what could be an algorithm that learnsthe embedded hierarchical structure?
Revised: Under this assumption, what could be an algorithm that learnsthe embedded hierarchical structure?

Original: We take inspiration from models in cognitive science showingthat people perceive structures based on the Gestalt principle of grouping by proximity , and formulatea generic hierarchical pattern discovery algorithm that enables the rational discovery of structureswith embedded hierarchies.
Revised: We take inspiration from models in cognitive science showingthat people perceive structures based on the Gestalt principle of grouping by proximity , and formulatea generic hierarchical pattern discovery algorithm that enables the rational discovery of structureswith embedded hierarchies.

Original: We refer to this as the hierarchical chunking model (HCM).
Revised: We refer to this as the hierarchical chunking model (HCM).

Original: HCM starts out learning a minimal set of units sufficient to explain the sequence and graduallycombines these units into increasingly larger and more complex chunks, constructing interpretablehierarchical structures.
Revised: HCM starts out learning a minimal set of units sufficient to explain the sequence and graduallycombines these units into increasingly larger and more complex chunks, constructing interpretablehierarchical structures.

Original: We derive learning guarantees on an idealized generative model and demonstrate convergence on sequential data coming from this generative model.
Revised: We derive learning guarantees on an idealized generative model and demonstrate convergence on sequential data coming from this generative model.

Original: Thereby, Gestalt principlesof grouping can be understood as a rational way of learning representations from sequences with aninherent hierarchical structure.
Revised: Thereby, Gestalt principlesof grouping can be understood as a rational way of learning representations from sequences with aninherent hierarchical structure.

Original: We then show that HCM resembles human learning in qualitativeways (compared to a recurrent neural network): it resembles human chunk learning and reactiontime measures and flexibly transfers components learned from one task to another.
Revised: We then show that HCM resembles human learning in qualitativeways (compared to a recurrent neural network): it resembles human chunk learning and reactiontime measures and flexibly transfers components learned from one task to another.

Original: We extended
Revised: We extended

Original: HCM to the visual-temporal domain capable of learning visual-temporal parts and wholes fromhigher dimensional sequential data.
Revised: HCM to the visual-temporal domain capable of learning visual-temporal parts and wholes fromhigher dimensional sequential data.

Original: Taking it one step further, we deploy HCM to learn from highdimensional fMRI data, which exerts a hierarchical structure.
Revised: Taking it one step further, we deploy HCM to learn from highdimensional fMRI data, which exerts a hierarchical structure.

Original: We demonstrate HCM’s interpretablefeature extraction ability to discover submodules of brain activations directly linkable to behaviorconfirmed by the neuroscientific literature and discoveries such as the correlation between brainchunk size with age and stimulus-specific brain chunks.
Revised: We demonstrate HCM’s interpretablefeature extraction ability to discover submodules of brain activations directly linkable to behaviorconfirmed by the neuroscientific literature and discoveries such as the correlation between brainchunk size with age and stimulus-specific brain chunks.

Original:  Hierarchical Chunking Model
Revised:  Hierarchical Chunking Model

Original: A chunk is a unit created by concatenating several atomic sequential units together.
Revised: A chunk is a unit created by concatenating several atomic sequential units together.

Original: Taking thetraining sequence shown in Figure 1a as an example, the sequence is made up of discrete, size-oneatomic units from an atomic alphabet A 0 : in this case A 0 = { 0 , 1 , 2 } .
Revised: Taking thetraining sequence shown in Figure 1a as an example, the sequence is made up of discrete, size-oneatomic units from an atomic alphabet A 0 : in this case A 0 = { 0 , 1 , 2 } .

Original: A chunk c is made up of acombination of one or more atomic units in A 0 \{ 0 } .
Revised: A chunk c is made up of acombination of one or more atomic units in A 0 \{ 0 } .

Original: 0 denotes an empty observation in the sequence.
Revised: 0 denotes an empty observation in the sequence.

Original: Intuitively, if a sequence contains inherent hierarchical structure, then there are patterns which spanseveral sequential units sharing these internal structures, examples of such sequences are repeatedmelodies and sub-melodies in music.
Revised: Intuitively, if a sequence contains inherent hierarchical structure, then there are patterns which spanseveral sequential units sharing these internal structures, examples of such sequences are repeatedmelodies and sub-melodies in music.

Original: If the pattern occurs in the sequence, observations betweensequential units within the pattern will be correlated.
Revised: If the pattern occurs in the sequence, observations betweensequential units within the pattern will be correlated.

Original: In this case, chunking patterns within a sequenceas units simplifies perceptual processing in the sense that the sequence can be perceived one chunkafter another, instead of one sequential unit at a time.
Revised: In this case, chunking patterns within a sequenceas units simplifies perceptual processing in the sense that the sequence can be perceived one chunkafter another, instead of one sequential unit at a time.

Original: Furthermore, the acquired “primary” chunksserve as building blocks to discover larger chunks that are embedded within the hierarchy of thesequential structure.
Revised: Furthermore, the acquired “primary” chunksserve as building blocks to discover larger chunks that are embedded within the hierarchy of thesequential structure.

Original: Formally, HCM acquires a belief set B of chunks, and uses chunks from the belief set to parse thesequence.
Revised: Formally, HCM acquires a belief set B of chunks, and uses chunks from the belief set to parse thesequence.

Original: HCM assumes that a sequence is generated from samples of independently occurringchunks with probability of P B ( c ) evaluated on the belief set B .
Revised: HCM assumes that a sequence is generated from samples of independently occurringchunks with probability of P B ( c ) evaluated on the belief set B .

Original: The probability of observinga sequence of parsed chunks c 1 , c 2 , ..., c N can be denoted as P ( c 1 , c 2 , ..., c N ) =
Revised: The probability of observinga sequence of parsed chunks c 1 , c 2 , ..., c N can be denoted as P ( c 1 , c 2 , ..., c N ) =

Original: (cid:81)
Revised: (cid:81)

Original: c
Revised: c

Original: i ∈ B P B ( c i ) .
Revised: i ∈ B P B ( c i ) .

Original: Chunks as perceiving units serve as independent factors that disentangle observations in the sequence.
Revised: Chunks as perceiving units serve as independent factors that disentangle observations in the sequence.

Original: Starting at the beginning of the sequence, HCM ‘perceives’ sequential units one chunk at a time,in other words, the training sequence is parsed by HCM into chunks.
Revised: Starting at the beginning of the sequence, HCM ‘perceives’ sequential units one chunk at a time,in other words, the training sequence is parsed by HCM into chunks.

Original: At every parsing step, thelongest chunk in the belief set that is consistent with the upcoming sequence is chosen to explain theup-coming sequential observations.
Revised: At every parsing step, thelongest chunk in the belief set that is consistent with the upcoming sequence is chosen to explain theup-coming sequential observations.

Original: The end of the previous chunk parse initiates the next parse.
Revised: The end of the previous chunk parse initiates the next parse.

Original: Observing a hierarchically structured sequence as illustrated in Figure 1a, HCM gradually builds upa hierarchy of chunks starting from an empty belief set.
Revised: Observing a hierarchically structured sequence as illustrated in Figure 1a, HCM gradually builds upa hierarchy of chunks starting from an empty belief set.

Original: It first identifies a set of atomic chunks toconstruct its initial belief set B .
Revised: It first identifies a set of atomic chunks toconstruct its initial belief set B .

Original: Initially, these will be chunks of length one, yielding one-by-oneprocessing of the primitive elements.
Revised: Initially, these will be chunks of length one, yielding one-by-oneprocessing of the primitive elements.

Original: For one belief set B , HCM keeps track of the marginal parsing frequency M ( c i ) for each chunk c i in 86 B , a vector with size | B | and the transition frequency T between chunk c i followed by chunk c j , asillustrated in Figure 1b.
Revised: For one belief set B , HCM keeps track of the marginal parsing frequency M ( c i ) for each chunk c i in 86 B , a vector with size | B | and the transition frequency T between chunk c i followed by chunk c j , asillustrated in Figure 1b.

Original: Entries in M and T are used to test the hypothesis that consecutive chunkparses have a correlated consecutive occurrence within the sequence via a χ 2 -independence test.
Revised: Entries in M and T are used to test the hypothesis that consecutive chunkparses have a correlated consecutive occurrence within the sequence via a χ 2 -independence test.

Original: If two chunks c L and c R have a significant adjacency dependence based on their entries in M and T ,they are chunked together to become c L ⊕ c R , which augments the belief set B by one.
Revised: If two chunks c L and c R have a significant adjacency dependence based on their entries in M and T ,they are chunked together to become c L ⊕ c R , which augments the belief set B by one.

Original: One exampleof chunk merging is shown in Figure 1c.
Revised: One exampleof chunk merging is shown in Figure 1c.

Original: Independence Test We use a χ 2 -test of independence to assess the correlation of consecutive occurrences of c L followed by c R in B .
Revised: Independence Test We use a χ 2 -test of independence to assess the correlation of consecutive occurrences of c L followed by c R in B .

Original: Let c L be an indicator variable that is 1 when chunk c l is parsed and 0 otherwise, similarly we formulate c R as another indicator variable of parsing the chunk c r .
Revised: Let c L be an indicator variable that is 1 when chunk c l is parsed and 0 otherwise, similarly we formulate c R as another indicator variable of parsing the chunk c r .

Original: We evaluate the χ 2 -value as a criterion to reject the null hypothesis that the consecutive observation of c l followed by c r is statistically independent:
Revised: We evaluate the χ 2 -value as a criterion to reject the null hypothesis that the consecutive observation of c l followed by c r is statistically independent:

Original: [Equation]
Revised: [Equation]

Original: p ( c L , c R ) and p ( c L ) p ( c R ) are evaluated from M an T .
Revised: p ( c L , c R ) and p ( c L ) p ( c R ) are evaluated from M an T .

Original: The degree of freedom is 1.
Revised: The degree of freedom is 1.

Original: A χ 2 -probabilityof less than 0.05 is the criterion to reject the null hypothesis (i.e. that c l and c r occur independently).
Revised: A χ 2 -probabilityof less than 0.05 is the criterion to reject the null hypothesis (i.e. that c l and c r occur independently).

Original: There are two versions of HCM.
Revised: There are two versions of HCM.

Original: The Rational Chunk Learning HCM learns chunks in an idealizedway which we use to study learning guarantees.
Revised: The Rational Chunk Learning HCM learns chunks in an idealizedway which we use to study learning guarantees.

Original: The online version of HCM is an approximationto the rational HCM that can be adapted to different environments and processes sequences online.
Revised: The online version of HCM is an approximationto the rational HCM that can be adapted to different environments and processes sequences online.

Original: Pseudo-code for both algorithms can be found in the SI.
Revised: Pseudo-code for both algorithms can be found in the SI.

Original: Rational Chunk Learning: HCM as an Ideal Observer HCM is initiated with an empty beliefset and it first finds a minimally complete belief set after the first sequence parse.
Revised: Rational Chunk Learning: HCM as an Ideal Observer HCM is initiated with an empty beliefset and it first finds a minimally complete belief set after the first sequence parse.

Original: In each iteration,the entire sequence is parsed to evaluate M and T , which are used to find consecutive chunk parsesin the existing belief set that violate the independence testing criterion.
Revised: In each iteration,the entire sequence is parsed to evaluate M and T , which are used to find consecutive chunk parsesin the existing belief set that violate the independence testing criterion.

Original: From these dependent chunkpairs, the pair with the largest estimated joint probability is combined into a new chunk.
Revised: From these dependent chunkpairs, the pair with the largest estimated joint probability is combined into a new chunk.

Original: The newchunk enlarges the belief set by one.
Revised: The newchunk enlarges the belief set by one.

Original: The chunks in the new belief set are used to parse the sequencein the next iteration.
Revised: The chunks in the new belief set are used to parse the sequencein the next iteration.

Original: This process repeats until all of the chunks in the belief set pass an independencehalting criterion, which measures if all of the chunks in the belief set are currently independent, againassessed via a 2 -test (see SI).
Revised: This process repeats until all of the chunks in the belief set pass an independencehalting criterion, which measures if all of the chunks in the belief set are currently independent, againassessed via a 2 -test (see SI).

Original: Online Chunk Learning The online chunk learning HCM approximates the ideal observer HCMby learning new chunks when the training sequence is processed online.
Revised: Online Chunk Learning The online chunk learning HCM approximates the ideal observer HCMby learning new chunks when the training sequence is processed online.

Original: To have a feature thatencourages adaptation to new environmental statistics, entries in M and T can be subject to memorydecay.
Revised: To have a feature thatencourages adaptation to new environmental statistics, entries in M and T can be subject to memorydecay.

Original: We use the ideal observer model to demonstrate learning guarantees, but use the online modelto learn representations in realistic and more complex set-ups.
Revised: We use the ideal observer model to demonstrate learning guarantees, but use the online modelto learn representations in realistic and more complex set-ups.

Original: 2.1 HCM Learns Representations from the Ground Up
Revised: 2.1 HCM Learns Representations from the Ground Up

Original: As HCM learns from a sequence, it starts with no representation and gradually builds up interpretablerepresentations described by a chunk hierarchy graph ˆ G with the vertex set being the chunks andedges pointing from constituents to composites.
Revised: As HCM learns from a sequence, it starts with no representation and gradually builds up interpretablerepresentations described by a chunk hierarchy graph ˆ G with the vertex set being the chunks andedges pointing from constituents to composites.

Original: Shown in Figure 1d is the gradual build-up of onesuch graph as the model learns from a training sequence coming from the generative hierarchy in
Revised: Shown in Figure 1d is the gradual build-up of onesuch graph as the model learns from a training sequence coming from the generative hierarchy in

Original: Figure 1a.
Revised: Figure 1a.

Original: At t = 10 , HCM learns only the atomic chunks, at t = 60 , HCM has already constructedtwo additional chunks; when t = 100 , two more additional chunks are constructed based on previousconstructions.
Revised: At t = 10 , HCM learns only the atomic chunks, at t = 60 , HCM has already constructedtwo additional chunks; when t = 100 , two more additional chunks are constructed based on previousconstructions.

Original: HCM arrives at the final chunk hierarchy at t = 150 .
Revised: HCM arrives at the final chunk hierarchy at t = 150 .

Original:  Generating Sequences with a Hierarchical Structure
Revised:  Generating Sequences with a Hierarchical Structure

Original: We construct a generative model to study HCM’s behavior formally and empirically.
Revised: We construct a generative model to study HCM’s behavior formally and empirically.

Original: The generativemodel constructs random chunk hierarchies from which non-iid sequences are sampled.
Revised: The generativemodel constructs random chunk hierarchies from which non-iid sequences are sampled.

Original: Such graph
Revised: Such graph

Original: G d contains vertex set V A d and edge set E A d to describe the relation between chunks and theirconstituents.
Revised: G d contains vertex set V A d and edge set E A d to describe the relation between chunks and theirconstituents.

Original: One example is illustrated in Figure 1a.
Revised: One example is illustrated in Figure 1a.

Original: A d is the set of chunks used to construct thesequence.
Revised: A d is the set of chunks used to construct thesequence.

Original: The depth d specifies the number of chunks created in the generative process.
Revised: The depth d specifies the number of chunks created in the generative process.

Original: Starting with an initial set of atomic chunks A 0 , at the i-th iteration, two chunks c L , c
Revised: Starting with an initial set of atomic chunks A 0 , at the i-th iteration, two chunks c L , c

Original: R are randomlychosen from the current set of chunks A i and are concatenated into a new chunk c L ⊕ c R , augmenting 128 A i by one to A i +1 .
Revised: R are randomlychosen from the current set of chunks A i and are concatenated into a new chunk c L ⊕ c R , augmenting 128 A i by one to A i +1 .

Original: Meanwhile, an independent occurrence probability is assigned to each chunkunder the constraint that the probability of occurrence for every new chunk c i in the constructionprocess evaluated on the support set A i carries the largest probability mass.
Revised: Meanwhile, an independent occurrence probability is assigned to each chunkunder the constraint that the probability of occurrence for every new chunk c i in the constructionprocess evaluated on the support set A i carries the largest probability mass.

Original: Once a graph hierarchy is constructed, we construct non-iid observational sequences by consecutivelysampling chunks from the hierarchy with their corresponding probability, under the constraint that notwo chunks with a child chunk are sampled consecutively.
Revised: Once a graph hierarchy is constructed, we construct non-iid observational sequences by consecutivelysampling chunks from the hierarchy with their corresponding probability, under the constraint that notwo chunks with a child chunk are sampled consecutively.

Original: 3.1 Learning Guarantee
Revised: 3.1 Learning Guarantee

Original: Theorem : As the length of the sequence approaches infinity, HCM learns a hierarchical chunkinggraph ˆ G isomorphic to the generative hierarchical graph G .
Revised: Theorem : As the length of the sequence approaches infinity, HCM learns a hierarchical chunkinggraph ˆ G isomorphic to the generative hierarchical graph G .

Original: Proof Sketch : We approach this proof by induction.
Revised: Proof Sketch : We approach this proof by induction.

Original: Further details can be found in the SI.
Revised: Further details can be found in the SI.

Original: Base step:
Revised: Base step:

Original: The first step of the rational chunking algorithm is to find the minimally complete atomic set of chunksto form its initial belief set.
Revised: The first step of the rational chunking algorithm is to find the minimally complete atomic set of chunksto form its initial belief set.

Original: This procedure guarantees that ˆ G 0 = G 0 .
Revised: This procedure guarantees that ˆ G 0 = G 0 .

Original: Additionally, the probabilitymass of the learning model at step 0 and the generative model at step 0 is asymptotically the sameas the sequence length approaches infinity.
Revised: Additionally, the probabilitymass of the learning model at step 0 and the generative model at step 0 is asymptotically the sameas the sequence length approaches infinity.

Original: Induction hypothesis: Assume that the learned beliefset B i at step i contains the same chunks as the alphabet set A i in the generative model, the chunkcombination pair with the biggest evaluated joint occurrence probability violating the independencetest is picked to be concatenated into a chunk to extend the belief set: this chunk is the same chunknode created by the hierarchical generative model.
Revised: Induction hypothesis: Assume that the learned beliefset B i at step i contains the same chunks as the alphabet set A i in the generative model, the chunkcombination pair with the biggest evaluated joint occurrence probability violating the independencetest is picked to be concatenated into a chunk to extend the belief set: this chunk is the same chunknode created by the hierarchical generative model.

Original: End step: The chunk learning process stops oncethe independence criterion is no longer violated.
Revised: End step: The chunk learning process stops oncethe independence criterion is no longer violated.

Original: This is the case once the chunk learning algorithmhas learned a belief set B d = A d .
Revised: This is the case once the chunk learning algorithmhas learned a belief set B d = A d .

Original: 3.2 Learning Convergence and Comparative Data-efficiency
Revised: 3.2 Learning Convergence and Comparative Data-efficiency

Original: To evaluate and show HCM’s learning performance, we trained HCM to learn hierarchies of chunksfrom sequences generated by the hierarchical generative model.
Revised: To evaluate and show HCM’s learning performance, we trained HCM to learn hierarchies of chunksfrom sequences generated by the hierarchical generative model.

Original: Shown in Figure 2 is HCM’s learningperformance as sequence length increases, averaged over 30 independently generated random graphswith the same depth d .
Revised: Shown in Figure 2 is HCM’s learningperformance as sequence length increases, averaged over 30 independently generated random graphswith the same depth d .

Original: One example of such graphs is shown in Figure 2a.
Revised: One example of such graphs is shown in Figure 2a.

Original: Kullback-Leiblerdivergence was used to evaluate learning performance.
Revised: Kullback-Leiblerdivergence was used to evaluate learning performance.

Original: To this end, learned hierarchies by HCM wereused to generate sequences, which were then evaluated on the support set of the alphabet set in thegenerative model.
Revised: To this end, learned hierarchies by HCM wereused to generate sequences, which were then evaluated on the support set of the alphabet set in thegenerative model.

Original: Figure 2b shows the KL-divergence between the learned and ground-truth distribution for increasingdepths d of the generative graphs.
Revised: Figure 2b shows the KL-divergence between the learned and ground-truth distribution for increasingdepths d of the generative graphs.

Original: For each depth, the KL-divergence was evaluated on 30 randomgenerative models with sequence length increasing from 50 to 3000.
Revised: For each depth, the KL-divergence was evaluated on 30 randomgenerative models with sequence length increasing from 50 to 3000.

Original: Overall, the KL-divergencedecreased as the training sequence length increased and converged with longer training sequences,showing a closer representation resemblance to the generative model.
Revised: Overall, the KL-divergencedecreased as the training sequence length increased and converged with longer training sequences,showing a closer representation resemblance to the generative model.

Original: A similar training and learning evaluation was conducted on a 3-layer Recurrent Neural Network(RNN) with 40 hidden units for comparison.
Revised: A similar training and learning evaluation was conducted on a 3-layer Recurrent Neural Network(RNN) with 40 hidden units for comparison.

Original: As the length of the training sequence increased, the
Revised: As the length of the training sequence increased, the

Original: KL-divergence of RNN converged at a slower rate than HCM.
Revised: KL-divergence of RNN converged at a slower rate than HCM.

Original: This competitive advantage in dataefficiency became more pronounced with increasing depth of the generative hierarchy.
Revised: This competitive advantage in dataefficiency became more pronounced with increasing depth of the generative hierarchy.

Original:  HCM Resembles Human Chunk Learning
Revised:  HCM Resembles Human Chunk Learning

Original: [Figure]
Revised: [Figure]

Original: Here we compare the chunk learning behavior of HCM to the learning characteristics of humans.
Revised: Here we compare the chunk learning behavior of HCM to the learning characteristics of humans.

Original: Tothat end, we used data collected from a sequence learning study by [23] with 47 participants underthe license CC-BY 4.0.
Revised: Tothat end, we used data collected from a sequence learning study by [23] with 47 participants underthe license CC-BY 4.0.

Original: As shown in Figure 3a, the training sequence comprised chunks ABC and D,independently occurring with equal probability.
Revised: As shown in Figure 3a, the training sequence comprised chunks ABC and D,independently occurring with equal probability.

Original: The study assessed how participants built up chunkknowledge gradually.
Revised: The study assessed how participants built up chunkknowledge gradually.

Original: Participants’ reaction times reflected that, after enough training, they wereanticipating several upcoming sequence elements, suggesting that they have acquired longer chunks(Figure 3b, left) [23].
Revised: Participants’ reaction times reflected that, after enough training, they wereanticipating several upcoming sequence elements, suggesting that they have acquired longer chunks(Figure 3b, left) [23].

Original: In a similar vein, we measured online chunk size increase of HCM using the same method asin [23] and, for comparison, RNNs (see SI for further comparisons to other algorithms).
Revised: In a similar vein, we measured online chunk size increase of HCM using the same method asin [23] and, for comparison, RNNs (see SI for further comparisons to other algorithms).

Original: HCM,similarly to humans, started learning longer chunks early in the sequence.
Revised: HCM,similarly to humans, started learning longer chunks early in the sequence.

Original: By contrast, RNN didnot start to chunk until after step 300, and when it started to learn chunks, the increase rate of thepredictive horizon was not as steep as HCM’s.
Revised: By contrast, RNN didnot start to chunk until after step 300, and when it started to learn chunks, the increase rate of thepredictive horizon was not as steep as HCM’s.

Original: Evaluating the average rate of chunk growth alsoshowed that HCM builds up chunks as learning progresses was more similar to participants’ thanthe RNN’s
Revised: Evaluating the average rate of chunk growth alsoshowed that HCM builds up chunks as learning progresses was more similar to participants’ thanthe RNN’s

Original: (Figure 3c).
Revised: (Figure 3c).

Original: The negative log-probabilities of sequence elements generated by the HCMand RNN were both significantly related to human reaction times (that reflect the certainty of theirinternal predictions [24]).
Revised: The negative log-probabilities of sequence elements generated by the HCMand RNN were both significantly related to human reaction times (that reflect the certainty of theirinternal predictions [24]).

Original: Yet, the relationship was substantially stronger (Figure 3d) between HCM( β = 16 .
Revised: Yet, the relationship was substantially stronger (Figure 3d) between HCM( β = 16 .

Original: 74 , p ≤ 0 . 001 ) and human participants compared to that of the RNN ( β = 9 .
Revised: 74 , p ≤ 0 . 001 ) and human participants compared to that of the RNN ( β = 9 .

Original: 24 , p ≤ 0 .
Revised: 24 , p ≤ 0 .

Original: These results suggest that HCM resembles human chunk learning more strongly than RNNs and cantherefore be seen as the cognitively more plausible approach to hierarchical representation learning.
Revised: These results suggest that HCM resembles human chunk learning more strongly than RNNs and cantherefore be seen as the cognitively more plausible approach to hierarchical representation learning.

Original:  HCM Permits Transfer Between Environments
Revised:  HCM Permits Transfer Between Environments

Original: One characteristic of human learning is that previous learning experience facilitates and sometimesinterferes with acquiring a new skill [25].
Revised: One characteristic of human learning is that previous learning experience facilitates and sometimesinterferes with acquiring a new skill [25].

Original: Having an interpretable representation can inform us aboutpositive or negative transfer a priori.
Revised: Having an interpretable representation can inform us aboutpositive or negative transfer a priori.

Original: An HCM has learned a chunking graph in Figure 4a.
Revised: An HCM has learned a chunking graph in Figure 4a.

Original: When it switches to another environmentwith overlapping representation, it can reuse the learned subgraphs of chunks marked in gray as in
Revised: When it switches to another environmentwith overlapping representation, it can reuse the learned subgraphs of chunks marked in gray as in

Original: Figure 4b and learns faster than a naive HCM in Figure 4c.
Revised: Figure 4b and learns faster than a naive HCM in Figure 4c.

Original: Vice versa, transfer can be detrimentalwhen there is no or little overlap between the learned chunking graph and the new environment.
Revised: Vice versa, transfer can be detrimentalwhen there is no or little overlap between the learned chunking graph and the new environment.

Original: Forthe same HCM as in Figure 4a, transferring to an environment with a chunking graph in Figure 4dimplies learning the shaded chunks anew, in addition to running the risk of being misled by theprevious representations.
Revised: Forthe same HCM as in Figure 4a, transferring to an environment with a chunking graph in Figure 4dimplies learning the shaded chunks anew, in addition to running the risk of being misled by theprevious representations.

Original: As a result, the early performance of the pre-trained HCM suffers more froman interfering environment than a naive model in Figure 4e.
Revised: As a result, the early performance of the pre-trained HCM suffers more froman interfering environment than a naive model in Figure 4e.

Original: Interpretability of HCM’s representationsenables the assessment of facilitation or interference when the environment changes.
Revised: Interpretability of HCM’s representationsenables the assessment of facilitation or interference when the environment changes.

Original:  Generalizing to Visual Temporal Chunks via the Principle of ProximalGrouping
Revised:  Generalizing to Visual Temporal Chunks via the Principle of ProximalGrouping

Original: Humans excel at finding structures in hierarchical visual objects and grouped movements.
Revised: Humans excel at finding structures in hierarchical visual objects and grouped movements.

Original: The Gestaltprinciple of grouping by proximity suggests that we tend to group objects that are close to one anotherinto a cohesive unit [26, 27].
Revised: The Gestaltprinciple of grouping by proximity suggests that we tend to group objects that are close to one anotherinto a cohesive unit [26, 27].

Original: This principle has been suggested to play a key role in human perceptualgrouping [28], benefits working memory [29] and reduces visual complexity [30].
Revised: This principle has been suggested to play a key role in human perceptualgrouping [28], benefits working memory [29] and reduces visual complexity [30].

Original: Indeed, in humansand other animals, learning of adjacent relationships prevails over non-adjacent ones [31].
Revised: Indeed, in humansand other animals, learning of adjacent relationships prevails over non-adjacent ones [31].

Original: Therefore,the adjacent dependency structure can be expanded to chunking in visual temporal domains [32].
Revised: Therefore,the adjacent dependency structure can be expanded to chunking in visual temporal domains [32].

Original: Toemulate this ability of chunking via proximal grouping, we extend HCM to learn visual temporalchunks.
Revised: Toemulate this ability of chunking via proximal grouping, we extend HCM to learn visual temporalchunks.

Original: Visual temporal chunks subsume temporal length and varying visual slices in each temporal
Revised: Visual temporal chunks subsume temporal length and varying visual slices in each temporal

Original: slice(Figure 5a).
Revised: slice(Figure 5a).

Original: One can imagine a visual temporal chunk as having a 3D shape — the first twodimensions are the visual part of the chunk, and the object’s length is the temporal part, made ofstacked visual-temporal pixels.
Revised: One can imagine a visual temporal chunk as having a 3D shape — the first twodimensions are the visual part of the chunk, and the object’s length is the temporal part, made ofstacked visual-temporal pixels.

Original: Within each temporal slice are the visual features identified bythe chunk.
Revised: Within each temporal slice are the visual features identified bythe chunk.

Original: As the model iterates through data across its temporal slice, the chunk that attains thebiggest visual temporal volume explains part of the observational sequence.
Revised: As the model iterates through data across its temporal slice, the chunk that attains thebiggest visual temporal volume explains part of the observational sequence.

Original: Multiple visual temporalchunks can occur simultaneously.
Revised: Multiple visual temporalchunks can occur simultaneously.

Original: Starting at the visual temporal time point marked by the previouschunk, chunks are identified and stored in M .
Revised: Starting at the visual temporal time point marked by the previouschunk, chunks are identified and stored in M .

Original: The transition matrix T is modified to account forthe temporal lag difference between adjacent chunk pairs within a proximity parameter and recordsthe frequency that one chunk transitions into another for each time lag.
Revised: The transition matrix T is modified to account forthe temporal lag difference between adjacent chunk pairs within a proximity parameter and recordsthe frequency that one chunk transitions into another for each time lag.

Original: Whenever a pair of adjacentchunks are identified, an independence hypothesis test evaluates whether the adjacent observation arecorrelated.
Revised: Whenever a pair of adjacentchunks are identified, an independence hypothesis test evaluates whether the adjacent observation arecorrelated.

Original: Chunks that violate the hypothesis test are combined to parse future sequences.
Revised: Chunks that violate the hypothesis test are combined to parse future sequences.

Original: Learning Part-Whole Relationship Between Visual Components We show HCM learned chunksin the visual domain from a sequence of independently sampled images.
Revised: Learning Part-Whole Relationship Between Visual Components We show HCM learned chunksin the visual domain from a sequence of independently sampled images.

Original: Figure 5b left shows ahierarchical generative model in the pixel-wise image domain.
Revised: Figure 5b left shows ahierarchical generative model in the pixel-wise image domain.

Original: A set of elementary visual units in the lowest hierarchy level combines into intermediate and more complex visual units higher up inthe hierarchy.
Revised: A set of elementary visual units inthe lowest hierarchy level combines into intermediate and more complex visual units higher up inthe hierarchy.

Original: All of the constructed elements in the hierarchy occurred independently according toa probability drawn from a Dirichlet flat distribution.
Revised: All of the constructed elements in the hierarchy occurred independently according toa probability drawn from a Dirichlet flat distribution.

Original: Images in the hierarchy were independentlysampled from the generative distribution to become the training sequence.
Revised: Images in the hierarchy were independently sampled from the generative distribution to become the training sequence.

Original: In Figure 5b, right weshow the chunk representations learned by HCM at different stages.
Revised: In Figure 5b, right weshow the chunk representations learned by HCM at different stages.

Original: Initially, HCM acquires theindividual pixels as chunks to explain the observations.
Revised: Initially, HCM acquires theindividual pixels as chunks to explain the observations.

Original: As HCM proceeds with learning, it discoversvisual correlations among the pixels and constructs increasingly complex visual parts.
Revised: As HCM proceeds with learning, it discoversvisual correlations among the pixels and constructs increasingly complex visual parts.

Original: Learning Visual-Temporal Movement Hierarchies Instead of seeing one image after anothersampled from an independent, identically distributed distribution, real-world experiences containcorrelations in both the visual and temporal dimensions.
Revised: Learning Visual-Temporal Movement Hierarchies Instead of seeing one image after anothersampled from an independent, identically distributed distribution, real-world experiences containcorrelations in both the visual and temporal dimensions.

Original: From observing object movements acrossspace and time, the visual system learns structures from correlated visual and temporal observations,decomposes motion structure, and groups moving objects together as a whole [33].
Revised: From observing object movements acrossspace and time, the visual system learns structures from correlated visual and temporal observations,decomposes motion structure, and groups moving objects together as a whole [33].

Original: To emulate thistype of environment, an animated GIF of a squid swimming in the sea (Figure 5c) was used as avisual-temporal sequence to train HCM.
Revised: To emulate thistype of environment, an animated GIF of a squid swimming in the sea (Figure 5c) was used as avisual-temporal sequence to train HCM.

Original: As learning advances, HCM learns chunks spanning boththe visual and temporal domains.
Revised: As learning advances, HCM learns chunks spanning boththe visual and temporal domains.

Original: There are visual-temporal chunks that mark the movements of atentacle and the rising-up motion of a bubble.
Revised: There are visual-temporal chunks that mark the movements of atentacle and the rising-up motion of a bubble.

Original: Additionally, visual chunks resemble a part of thevisual’s eye and face.
Revised: Additionally, visual chunks resemble a part of thevisual’s eye and face.

Original: The meaningful chunks in the visual-temporal domain suggest the groupingprinciple enables the plausible learning of movement sequences and aids the perception of objects aswholes and their corresponding parts.
Revised: The meaningful chunks in the visual-temporal domain suggest the groupingprinciple enables the plausible learning of movement sequences and aids the perception of objects aswholes and their corresponding parts.

Original:  Learning Hierarchies of Brain Activation from Resting-state fMRI data
Revised:  Learning Hierarchies of Brain Activation from Resting-state fMRI data

Original: HCM learns hierarchies from structured sequential data.
Revised: HCM learns hierarchies from structured sequential data.

Original: As brain activation has been suggested to behierarchically structured [34], we demonstrate HCM’s usefulness to learn structures in biologicalneural networks activating in response to complex stimuli by running HCM on a resting-state fMRIdata set.
Revised: As brain activation has been suggested to behierarchically structured [34], we demonstrate HCM’s usefulness to learn structures in biologicalneural networks activating in response to complex stimuli by running HCM on a resting-state fMRIdata set.

Original: We used a developmental data set provided by the nilearn package with BSD License [35] andoriginally collected by [36].
Revised: We used a developmental data set provided by the nilearn package with BSD License [35] andoriginally collected by [36].

Original: This data set contains the resting-state BOLD activity of 155 participantsranging from age 3 to 40, while watching the silent movie “Partly Cloudy”.
Revised: This data set contains the resting-state BOLD activity of 155 participantsranging from age 3 to 40, while watching the silent movie “Partly Cloudy”.

Original: BOLD signal wasextracted from functional brain regions defined by the MSDL Atlas [37], with confounds excludedand transformed into a rounded, normalized time series.
Revised: BOLD signal wasextracted from functional brain regions defined by the MSDL Atlas [37], with confounds excludedand transformed into a rounded, normalized time series.

Original: HCM’s Chunks Reflect Structural, Functional and Anatomical Connectivity Figure 6 showsthree typical examples of learned chunks for a randomly-chosen participant.
Revised: HCM’s Chunks Reflect Structural, Functional and Anatomical Connectivity Figure 6 showsthree typical examples of learned chunks for a randomly-chosen participant.

Original: The labels of functionalregions come from the MSDL atlas [37].
Revised: The labels of functionalregions come from the MSDL atlas [37].

Original: The first example is the co-activation of D ACC and R A Ins.
Revised: The first example is the co-activation of D ACC and R A Ins.

Original: These two regions have been observed to co-activate in the presence of emotions, pain, and humor.
Revised: These two regions have been observed to co-activate in the presence of emotions, pain, and humor.

Original: They have been suggested to be a key hub of the salience network [38–41].
Revised: They have been suggested to be a key hub of the salience network [38–41].

Original: The second example chunk contains the activation of R DLPFC and R Front Pole.
Revised: The second examplechunk contains the activation of R DLPFC and R Front Pole.

Original: These regions belong to the visualattention network and are known to be anatomically connected [42–44].
Revised: These regions belong to the visualattention network and are known to be anatomically connected [42–44].

Original: A final example is the chunkof L Ins and Cing, which are also known to be anatomically and functionally connected [45].
Revised: A final example is the chunk of L Ins and Cing, which are also known to be anatomically and functionally connected [45].

Original: Thus,the chunks discovered by HCM correspond to empirically-verified patterns of functional activity.
Revised: Thus,the chunks discovered by HCM correspond to empirically-verified patterns of functional activity.

Original: HCM’s Chunks Recover Hierarchical Activation Patterns In fMRI data, hierarchies of chunkactivation constructed by HCM reflect networks of functional regions.
Revised: HCM’s Chunks Recover Hierarchical Activation Patterns In fMRI data, hierarchies of chunkactivation constructed by HCM reflect networks of functional regions.

Original: On the top of the hierarchy,the largest chunk contained L DLPFC, D ACC, R A Ins, Cing, and L Ant IPS (Figure 6b).
Revised: On the top of the hierarchy,the largest chunk contained L DLPFC, D ACC, R A Ins, Cing, and L Ant IPS (Figure 6b).

Original: Thoseregions are known to co-activate during cognitive tasks that demand attention, working memory, andcontrol [41].
Revised: Thoseregions are known to co-activate during cognitive tasks that demand attention, working memory, andcontrol [41].

Original: Chunks in the intermediate levels of the hierarchy reflect sub-networks of functionalconnectivity.
Revised: Chunks in the intermediate levels of the hierarchy reflect sub-networks of functionalconnectivity.

Original: Sub-chunks such as D ACC, R A Ins, L Ant IPS, and L DLPFC have been suggested toconjointly activate in cognitive effort-related activities [41].
Revised: Sub-chunks such as D ACC, R A Ins, L Ant IPS, and L DLPFC have been suggested toconjointly activate in cognitive effort-related activities [41].

Original: Atomic chunks in the hierarchy suchas D ACC and L Ant IPS activate individually sometimes without their parent chunks.
Revised: Atomic chunks in the hierarchy suchas D ACC and L Ant IPS activate individually sometimes without their parent chunks.

Original: Indeed, theyhave distinct functional signatures for affect processing [46, 47], and visual attention control [48].
Revised: Indeed, theyhave distinct functional signatures for affect processing [46, 47], and visual attention control [48].

Original: Upon exposure to a time series in fMRI data, HCM constructs chunks from their constituents andarrives at a hierarchy of chunk relations, indicating nested network structures in the human brain.
Revised: Upon exposure to a time series in fMRI data, HCM constructs chunks from their constituents andarrives at a hierarchy of chunk relations, indicating nested network structures in the human brain.

Original: HCM’s Chunks can be Matched with Stimulus Onsets
Revised: HCM’s Chunks can be Matched with Stimulus Onsets

Original: The retrieved chunks by HCM can betagged with critical stimulus onsets.
Revised: The retrieved chunks by HCM can betagged with critical stimulus onsets.

Original: We tagged 19 critical moments involving social and emotionalcontent in the movie.
Revised: We tagged 19 critical moments involving social and emotionalcontent in the movie.

Original: Figure 6c shows one example chunk activation upon stimulus onsets.
Revised: Figure 6c shows one example chunk activation upon stimulus onsets.

Original: Frontal
Revised: Frontal

Original: DMN, right frontal pole, ventral anterior cingulate cortex, and right anterior insula activate togetheras a recurring unit after participants witness a scene with characters greeting and hugging each other.
Revised: DMN, right frontal pole, ventral anterior cingulate cortex, and right anterior insula activate togetheras a recurring unit after participants witness a scene with characters greeting and hugging each other.

Original: These regions have been suggested to be involved in social and cognitive processing [49].
Revised: These regions have been suggested to be involved in social and cognitive processing [49].

Original: Anotherexample is the activation of areas known to be involved in emotion and language processing: D ACCand Broca [50], during a scene containing social interactions.
Revised: Anotherexample is the activation of areas known to be involved in emotion and language processing: D ACCand Broca [50], during a scene containing social interactions.

Original: In the meantime, the left and rightprefrontal cortex, involved in theory-of-mind [36], also lights up.
Revised: In the meantime, the left and rightprefrontal cortex, involved in theory-of-mind [36], also lights up.

Original: We categorized the tagged moments into 3 groups of different emotional load: sadness, anger, andcompassion.
Revised: We categorized the tagged moments into 3 groups of different emotional load: sadness, anger, andcompassion.

Original: We then looked at the activation probability of retrieved chunks within the 6 secondsafter watching those tagged scenes.
Revised: We then looked at the activation probability of retrieved chunks within the 6 secondsafter watching those tagged scenes.

Original: Figure 6d shows a list of such chunks from one participant withtheir activation probability for each emotional category.
Revised: Figure 6d shows a list of such chunks from one participant withtheir activation probability for each emotional category.

Original: For example, the left and right insula, knownto be involved in affective processing [51], have a 0.4 activation probability after witnessing scenesof sadness or anger, but no activation after witnessing scenes of compassion.
Revised: For example, the left and right insula, knownto be involved in affective processing [51], have a 0.4 activation probability after witnessing scenesof sadness or anger, but no activation after witnessing scenes of compassion.

Original: The same holds for R
Revised: The same holds for R

Original: DLPFC and R Par that have been documented to activate in response to emotional conflict [52].
Revised: DLPFC and R Par that have been documented to activate in response to emotional conflict [52].

Original: On the other hand, regions such as R TPJ and R pars opercularis that are involved in emotional reactionsand theory of mind processes [53], activate in response to a scene of compassion or anger, but notto a scene of sadness.
Revised: Onthe other hand, regions such as R TPJ and R pars opercularis that are involved in emotional reactionsand theory of mind processes [53], activate in response to a scene of compassion or anger, but not to a scene of sadness.

Original: Thus, chunks of active brain regions can be related to complex stimuli, andregions activate selectively in response to one or more categories of emotional stimuli, but not others.
Revised: Thus, chunks of active brain regions can be related to complex stimuli, andregions activate selectively in response to one or more categories of emotional stimuli, but not others.

Original: HCM’s Average Chunk size Correlates with Participants’ Age HCM can also be used to performmeaningful analyses at the population level.
Revised: HCM’s Average Chunk size Correlates with Participants’ Age HCM can also be used to performmeaningful analyses at the population level.

Original: Specifically, we find that HCM’s returned average chunksize per participant correlates significantly with age (Figure 6e).
Revised: Specifically, we find that HCM’s returned average chunksize per participant correlates significantly with age (Figure 6e).

Original: The older the participants are, thelonger are the chunks found in their data ( r = 0 .
Revised: The older the participants are, thelonger are the chunks found in their data ( r = 0 .

Original: 23 , p ≤ 0 . 001 ).
Revised: 23 , p ≤ 0 . 001 ).

Original: This discovery is in line withfindings in the original study, which showed an increase in modularization of ToM and pain circuitsacross development [36].
Revised: This discovery is in line withfindings in the original study, which showed an increase in modularization of ToM and pain circuitsacross development [36].

Original: To summarize, we applied HCM to learn chunks from a developmental fMRI data set.
Revised: To summarize, we applied HCM to learn chunks from a developmental fMRI data set.

Original: HCM enabledthe discovery of spatially and temporally correlated activation chunks that are theoretically andempirically meaningful.
Revised: HCM enabledthe discovery of spatially and temporally correlated activation chunks that are theoretically andempirically meaningful.

Original: The resulting chunks can be linked to complex stimuli and offer directlyinterpretable insights into the structure and function of brain activity.
Revised: The resulting chunks can be linked to complex stimuli and offer directlyinterpretable insights into the structure and function of brain activity.

Original:  Discussion
Revised:  Discussion

Original: HCM extends upon decades of previous cognitive science and psychology research on chunking.
Revised: HCM extends upon decades of previous cognitive science and psychology research on chunking.

Original: Incognitive science, process models such as PARSER and competitive chunking were demonstratedto generate qualitatively similar chunks as in human sequence learning [54, 55].
Revised: Incognitive science, process models such as PARSER and competitive chunking were demonstratedto generate qualitatively similar chunks as in human sequence learning [54, 55].

Original: HCM is a rationalalgorithm that learns the underlying chunks when the sequence is generated from a hierarchicalchunking graph.
Revised: HCM is a rationalalgorithm that learns the underlying chunks when the sequence is generated from a hierarchicalchunking graph.

Original: Therefore, the chunking criterion is no longer a heuristic but a rational learning strategy that enables hierarchical structural discovery.
Revised: Therefore, the chunking criterion is no longer a heuristic but a rational learning strategy that enables hierarchical structural discovery.

Original: On top of inheriting the merits of its predecessors,
Revised: On top of inheriting the merits of its predecessors,

Original: HCM generalizes the chunk learning principle to higher dimensional sequential domains such asvisual-temporal sequential data.
Revised: HCM generalizes the chunk learning principle to higher dimensional sequential domains such asvisual-temporal sequential data.

Original: Another related line of research is program induction, where explicit representations are acquired bysearching for programmatic structures that best explain observational samples [22], and consolidatingthese offline with library learning [56].
Revised: Another related line of research is program induction, where explicit representations are acquired bysearching for programmatic structures that best explain observational samples [22], and consolidatingthese offline with library learning [56].

Original: However, primitive programs need domain expert knowledgeto specify; the relations and composition rules must adapt to the task settings and sensitively influencethe quality of retrieved representations.
Revised: However, primitive programs need domain expert knowledgeto specify; the relations and composition rules must adapt to the task settings and sensitively influencethe quality of retrieved representations.

Original: The one-dimensional HCM is similar to natural languageprocessing methods such as Byte-Pair Encoding [57] or WordPiece [58], which were developed tooptimize sequential data compression instead of a cognitively inspired background and works withone-dimensional sequences.
Revised: The one-dimensional HCM is similar to natural languageprocessing methods such as Byte-Pair Encoding [57] or WordPiece [58], which were developed tooptimize sequential data compression instead of a cognitively inspired background and works withone-dimensional sequences.

Original: Other approaches to structure learning include unsupervised parsing[59], which learns a stochastic and-or graph from sequential data.
Revised: Other approaches to structure learning include unsupervised parsing[59], which learns a stochastic and-or graph from sequential data.

Original: HCM is distinct in adapting itsrepresentation granularity to discover bigger chunks from data without pre-specifying the structure.
Revised: HCM is distinct in adapting itsrepresentation granularity to discover bigger chunks from data without pre-specifying the structure.

Original: Our work has its limitations.
Revised: Our work has its limitations.

Original: At the moment, HCM learns representations from semi-high dimensionalsequential data.
Revised: At the moment, HCM learns representations from semi-high dimensionalsequential data.

Original: We are actively looking into generalizing this algorithm to higher dimensional datadomains by combining it with existing neural network approaches or computer vision algorithmssuch as coherent point drift [60] or normalized cuts [61] to allow for the learning of ambiguous andhigh dimensional chunk exemplars.
Revised: We are actively looking into generalizing this algorithm to higher dimensional datadomains by combining it with existing neural network approaches or computer vision algorithmssuch as coherent point drift [60] or normalized cuts [61] to allow for the learning of ambiguous andhigh dimensional chunk exemplars.

Original: Another limitation is that HCM learns one type of hierarchy ofcompound representations.
Revised: Another limitation is that HCM learns one type of hierarchy ofcompound representations.

Original: However, we can show that HCM can be generalized to not only learnsimple chunks but also chunks in projected spaces and thereby generalize between two chunks thatcontain the same motif (for example, “12221212” and “34443434”; see SI for detailed results).
Revised: However, we can show that HCM can be generalized to not only learnsimple chunks but also chunks in projected spaces and thereby generalize between two chunks thatcontain the same motif (for example, “12221212” and “34443434”; see SI for detailed results).

Original: In thefuture, it might be worthwhile to further combine our approach with others amongst a taxonomy ofrepresentational hierarchies.
Revised: In thefuture, it might be worthwhile to further combine our approach with others amongst a taxonomy ofrepresentational hierarchies.

Original: HCM also opens up other application directions.
Revised: HCM also opens up other application directions.

Original: One direction is integrating HCM with deep neuralnetwork approaches as an interface between human understanding and distributed computation.
Revised: One direction is integrating HCM with deep neuralnetwork approaches as an interface between human understanding and distributed computation.

Original: Learning hierarchies of coherent activations from intermediate hidden units has the potential to revealneural networks’ underlying computation structure.
Revised: Learning hierarchies of coherent activations from intermediate hidden units has the potential to revealneural networks’ underlying computation structure.

Original: Another direction in neuroscience or behavioralresearch is to learn chunks of tagged animal movements that enable insights into the emergence ofbehavioral structure [62].
Revised: Another direction in neuroscience or behavioralresearch is to learn chunks of tagged animal movements that enable insights into the emergence ofbehavioral structure [62].

Original: Finally, finding patterns that form as a cognitive unit is a vital task forinfants to learn about the structures of the world and resembles the process of formulating a scientifictheory from observation [63].
Revised: Finally, finding patterns that form as a cognitive unit is a vital task forinfants to learn about the structures of the world and resembles the process of formulating a scientifictheory from observation [63].

Original: HCM can function as one means to come up with theories of theworld by observation, ready for experimental interventions, or active learning to delineate the causalstructure within [64].
Revised: HCM can function as one means to come up with theories of theworld by observation, ready for experimental interventions, or active learning to delineate the causalstructure within [64].

Original: Checklist
Revised: Checklist

Original: For all authors...
Revised: For all authors...

Original: (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s
Revised: (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s

Original: contributions and scope?
Revised: contributions and scope?

Original: [Yes]
Revised: [Yes]

Original: (b) Did you describe the limitations of your work?
Revised: (b) Did you describe the limitations of your work?

Original: [Yes]
Revised: [Yes]

Original: (c) Did you discuss any potential negative societal impacts of your work?
Revised: (c) Did you discuss any potential negative societal impacts of your work?

Original: [N/A]
Revised: [N/A]

Original: As this
Revised: As this

Original: work introduces an algorithmic approach, it is not clear to us with the potential negative
Revised: work introduces an algorithmic approach, it is not clear to us with the potential negative

Original: societal impact it can induce.
Revised: societal impact it can induce.

Original: (d) Have you read the ethics review guidelines and ensured that your paper conforms tothem?
Revised: (d) Have you read the ethics review guidelines and ensured that your paper conforms tothem?

Original: [Yes]
Revised: [Yes]

Original: If you are including theoretical results...
Revised: If you are including theoretical results...

Original: (a) Did you state the full set of assumptions of all theoretical results?
Revised: (a) Did you state the full set of assumptions of all theoretical results?

Original: [Yes]
Revised: [Yes]

Original: (b) Did you include complete proofs of all theoretical results?
Revised: (b) Did you include complete proofs of all theoretical results?

Original: [Yes] A proof sketch is
Revised: [Yes] A proof sketch is

Original: described in the main paper, while the full proof can be found in the supplementary
Revised: described in the main paper, while the full proof can be found in the supplementary

Original: material.
Revised: material.

Original: If you ran experiments...
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Original: (a) Did you include the code, data, and instructions needed to reproduce the main experi-
Revised: (a) Did you include the code, data, and instructions needed to reproduce the main experi-

Original: mental results (either in the supplemental material or as a URL)?
Revised: mental results (either in the supplemental material or as a URL)?

Original: [Yes] They will beincluded in the supplementary material and publicly available.
Revised: [Yes] They will beincluded in the supplementary material and publicly available.

Original: (b) Did you specify all the training details (e.g., data splits, hyperparameters,
Revised: (b) Did you specify all the training details (e.g., data splits, hyperparameters,

Original: how they
Revised: how they

Original: were chosen)?
Revised: were chosen)?

Original: [Yes] Details on the experiments will be included in the supplementary.
Revised: [Yes] Details on the experiments will be included in the supplementary.

Original: (c) Did you report error bars (e.g., with respect to the random seed after running experi-
Revised: (c) Did you report error bars (e.g., with respect to the random seed after running experi-

Original: ments multiple times)?
Revised: ments multiple times)?

Original: [Yes]
Revised: [Yes]

Original: (d) Did you include the total amount of compute and the type of resources used (e.g., typeof GPUs, internal cluster, or cloud provider)?
Revised: (d) Did you include the total amount of compute and the type of resources used (e.g., typeof GPUs, internal cluster, or cloud provider)?

Original: [N/A] Results obtained run without
Revised: [N/A] Results obtained run without

Original: special computing resources.
Revised: special computing resources.

Original: If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
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Original: (a) If your work uses existing assets, did you cite the creators?
Revised: (a) If your work uses existing assets, did you cite the creators?

Original: [Yes]
Revised: [Yes]

Original: (b) Did you mention the license of the assets?
Revised: (b) Did you mention the license of the assets?

Original: [Yes]
Revised: [Yes]

Original: (c) Did you include any new assets either in the supplemental material or as a URL?
Revised: (c) Did you include any new assets either in the supplemental material or as a URL?

Original: [No]
Revised: [No]

Original: (d) Did you discuss whether and how consent was obtained from people whose data you’re
Revised: (d) Did you discuss whether and how consent was obtained from people whose data you’re

Original: using/curating?
Revised: using/curating?

Original: [Yes]
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Original: (e) Did you discuss whether the data you are using/curating contains personally identifiable
Revised: (e) Did you discuss whether the data you are using/curating contains personally identifiable

Original: information or offensive content?
Revised: information or offensive content?

Original: [No] From our awareness, no personally identifiable
Revised: [No] From our awareness, no personally identifiable

Original: information is present in any of the assets.
Revised: information is present in any of the assets.

Original: If you used crowdsourcing or conducted research with human participants...
Revised: If you used crowdsourcing or conducted research with human participants...

Original: (a) Did you include the full text of instructions given to participants and screenshots, if
Revised: (a) Did you include the full text of instructions given to participants and screenshots, if

Original: applicable?
Revised: applicable?

Original: [N/A] We used existing data from experiments conducted in previous
Revised: [N/A] We used existing data from experiments conducted in previous

Original: work.
Revised: work.

Original: This information can be found in the original articles.
Revised: This information can be found in the original articles.

Original: (b) Did you describe any potential participant risks, with links to Institutional Review
Revised: (b) Did you describe any potential participant risks, with links to Institutional Review

Original: Board (IRB) approvals, if applicable?
Revised: Board (IRB) approvals, if applicable?

Original: [N/A]
Revised: [N/A]

Original: This question is not applicable to this
Revised: This question is not applicable to this

Original: project.
Revised: project.

Original: (c) Did you include the estimated hourly wage paid to participants and the total amount
Revised: (c) Did you include the estimated hourly wage paid to participants and the total amount

Original: spent on participant compensation?
Revised: spent on participant compensation?

Original: [N/A] Since no human experiment is conducted for
Revised: [N/A] Since no human experiment is conducted for

Original: this project and we used experimental data from other published and publicly available
Revised: this project and we used experimental data from other published and publicly available

Original: work, this question is not applicable to us here and can be found in the original articles.
Revised: work, this question is not applicable to us here and can be found in the original articles.