Learning Algebraic Representation for Abstract Spatial-Temporal Reasoning
Authors: Chi Zhang
Sirui Xie
Baoxiong Jia
Yixin Zhu
Ying Nian Wu
Song-Chun Zhu
Venue:  (2021)

Abstract: Is intelligence realized by connectionist or classicist? While connectionist approaches have achieved superhuman performance, there has been growing evidence that such task-specific superiority is particularly fragile in systematic generalization. This observation lies in the central debate (Fodor et al., 1988; Fodor &McLaughlin, 1990) between connectionist and classicist, wherein the latter continually advocates an algebraic treatment in cognitive architectures. In this work, we follow the classicist's call and propose a hybrid approach to improve systematic generalization in reasoning. Specifically, we showcase a prototype with algebraic representations for the abstract spatial-temporal reasoning task of Raven’s Progressive Matrices (RPM) and present the ALgebra-Aware Neuro-Semi-Symbolic (ALANS$^2$) learner. The ALANS$^2$ learner is motivated by abstract algebra and the representation theory. It consists of a neural visual perception frontend and an algebraic abstract reasoning backend: the frontend summarizes the visual information from object-based representations, while the backend transforms it into an algebraic structure and induces the hidden operator on-the-fly. The induced operator is later executed to predict the answer's representation, and the choice most similar to the prediction is selected as the solution. Extensive experiments show that by incorporating an algebraic treatment, the ALANS$^2$ learner outperforms various pure connectionist models in domains requiring systematic generalization. We further show that the algebraic representation learned can be decoded by isomorphism and used to generate an answer.

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LEARNING ALGEBRAIC REPRESENTATION FOR ABSTRACT
REASONING
Anonymous authors Paper under double-blind review
ABSTRACT
Is intelligence realized by connectionist or classicist? While connectionist ap- proaches have achieved superhuman performance, there has been growing evi- dence that such task-specific superiority is particularly fragile in systematic gen- eralization. This observation lies in the central debate (Fodor et al.) 1988) Fodor & McLaughlin 1990) between connectionist and classicist, wherein the latter contin- ually advocates an algebraic treatment cognitive architectures. this work, we follow the classicist's s call and propose a hybrid approach to improve systematic generalization in reasoning. Specifically, we showease prototype with algebraic representations for the abstract spatial-temporal reasoning task of Raven's is Pro- gressive Matrices (RPM) and present the ALgebra-Aware Seuro-Semi-Symbee (ALANS²) learner. The ALANS* learner is motivated by abstract algebra and the representation theory. It consists of a neural visual perception frontend and an algebraic abstract reasoning backend: the frontend summarizes the visual infor- mation from object- object-based representations, while the backend transforms it into an algebraic structure and induces the hidden operator on-the-fly. The induced oper- ator is later executed predict the answer's representation, and the choice most similar to the prediction is selected as the solution. Extensive experiments show algebraic treatment, the ALANS² learner outperforms
that by incorporating various pure connectionist models in domains requiring systematic generaliza- tion. We further show that the algebraic representation learned can be decoded by isomorphism and used to generate an answer.
INTRODUCTION
"Thought is in fact kind of Algebra."
William James (Jaess 1891
Imagine you are given two alphabetical sequences of "c, b, a" and "d, c, b", and asked to fill in the missing element in "e, d, ?". In nearly no time will one realize the answer to be c. However, more surprising for human learning is that, effortlessly and instantaneously. we can "freely general- ize" (Marcus 2001) the solution to any partial consecutive ordered sequences. While believed to be innate in early development for human infants (Marcus et al., 1999), such systematic generalizability has constantly been missing and proven to be particularly challenging in existing connectionist mod- els (Lake & Baroni 2018 Bahdanau et al., 2019). In fact, such an ability to entertain given thought and semantically re related contents strongly implies an abstract algebra-like treatment (Fodor et al., 1988); in literature, it is referred to as the "language of thought" (Fodor 1975). "physical symbol system" " (Newell 1980), and "algebraic mind" (Marcus) (001]. However, in stark contrast, exist- ing connectionist models tend only to capture statistical correlation (Lake & Baroni 2018 Kansky al.) 2017 Chollet 2019), rather than providing any account for a structural inductive bias where systematic algebra can be carried out to facilitate generalization.
This contrast instinctively raises question-what constitutes such an algebraic inductive bias? We argue that the foundation of the modeling counterpart to the algebraic treatment early human development (Marcus 2001 Marcus et al. 1999) lies in algebraic computations set up on mathe- matical axioms, a form of formalized human intuition and the starting point of modern mathematical reasoning (Heath et al., 1956 Maddy 1988). Of particular importance to the basic building blocks of algebra is the Peano Axiom (Peano) 1889 In the Peano Axiom, the essential components of alge- bra, the algebraic set and corresponding operators over it. are governed by three statements: (1) the existence of at least one element in the field to study ("zero" element), (2) successor function that is recursively applied to all elements and can, therefore, span the entire field, and (3) the principle of

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mathematical induction. Building such fundamental axiom, we begin form the notion of an algebraic set and induce the operator along with to construct an algebraic structure. We hypothe- size that such a treatment of algebraic computations set up on fundamental axioms is essential for a model's systematic generalizability, the lack of which will only make it sub-optimal.
To demonstrate the benefits of such an algebraic treatment in systematic generalization, we show- case prototype for Raven's Progressive Matrices (RPM) (Raven) 1936; Raven & Court 1998), an exemplar task for abstract spatial- this task, an agent is given an incomplete 3 x 3 matrix consisting of eight context panels with the last one missing, and asked to pick one answer from a set of eight choices that best completes the matrix. Human's 's reasoning capability of solving this abstract reasoning task has been commonly regarded as an indicator of "general intelligence" (Carpenter et al. 1990)) and "fluid intelligence" " (Spearman Hofstadter 1995 Jaeggi et 2008). In spite of the task being one that ideally requires abstraction, algebraization, induction, and generalization (Raven 1936; Raven & Court 1998 Car- penter al. 1990). recent endeavors unanimously propose pure connectionist models that attempt to circumvent such intrinsic cognitive requirements (Santoro et al. 2018: Zhang et aL. 22119 bj Wang et al. 2020; Zheng et al.) 2019 Hu et al. 2020; Wu et 2020).. However, these methods ineffi ciency is also evident systematic generalization, they struggle to extrapolate to domains beyond training, as pointed out in (Santoro et al. 2018 Zhang et al. 2019b) and shown later in this paper.
reasoning (Santoro et al. 2018 2018: Zhang et al. (019a). In
1923 1927
Heuro-Semi-Symboli (ALANS?) learner. a high-leell, the ALANS learner is embedded in general neuro-symbolic architecture (Yi et al., 2018 Mao et al.) 2019 Han et al.] 2019 Yi et al. (0200 but has on- on-the-fly operator learnability and hence semi-symbolic. Specifically, consists of a neural visual perception frontend and an algebraic abstract reasoning backend. For each RPM instance, the neural visual perception frontend first slides window over each panel to obtain the object- based representations (Kansky al., 2017 [u Wu al., 2017) for every object. A belief inference engine latter aggregates all object-based representations in each panel to produce the probabilistic belief state. The algebraic abstract reasoning backend then takes the belief states of the eight context panels, treats them as snapshots on an algebraic structure, lifts them into a matrix-based algebraic representation built on the Peano Axiom and the representation theory (Humphreys. 2012). and induces the hidden operator in the algebraic structure by solving an inner optimization (Colson et al. 2007| [aard] 2013). The algebraic representation for the answer is predicted by executing the induced operator: its corresponding set element is decoded by isomorphism established in the representation theory, and the final answer is selected as the one most similar to the prediction.
To address the issue, we introduce the ALgebra-Aware
The ALANS learner enjoys several benefits in abstract reasoning with an algebraic treatment:
1. Unlike previous monolithic models, the ALANS² learner offers a more interpretable account of the entire abstract reasoning process: the neural visual perception frontend extracts object-based representations and produces belief states of panels by explicit probability inference, whereas the algebraic abstract reasoning backend induces the hidden operator in the algebraic structure. The corresponding representation for the final answer is obtained by executing the induced operator, and the choice panel with minimum distance is selected. This process much resembles the top- down bottom-up strategy in human reasoning: humans reason by inducing the hidden relation, executing it to generate feasible solution in mind, and choosing the most similar answer avail- able (Carpenter et al. 1990). Such a strategy is missing in recent literature (Santoro et al.) 2018 Zhhang et al. by Wang et al. 20200] Zheng et al. 2219 Hu et al. 2020 Wa et al. (022)).
While keeping the semantic interpretability and end-to-end trainability in existing neuro- symbolic frameworks (Yi et al.] 2018; Mao et al.) 2019 Han et al., 2019, Yi et al. 2020).
ALANS is what we call semi-symbolic
the sense that the symbolic operator can be learned
and concluded on-the-ff without manual definition for every one of them. Such an inductive ability also enables greater extent of the desired generalizability.
3. By decoding the predicted representation in the algebraic structure, we can also generate an answer that satisfies the hidden relation in the context.
This work makes three major contributions: (1) We propose the ALANS² learner. Compared to existing monolithic models, the ALANS2 learner adopts seuro-semi-symbolli design, where the problem-solving process is decomposed into neural visual perception and algebraic abstract rea- soning. (2) To demonstrate the efficacy of incorporating an algebraic treatment in abstract spatial- temporal reasoning, we show the superior systematic generalization ability of the proposed ALANS? leamnr in various extrapolatory RPM domains. (3) We present analyses into both neural visual per- ception and algebraic abstract reasoning. We also show the generative potential of ALANS
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2 RELATED WORK
Quest for Symbolized Manipulation The idea to treat thinking as mental language can be dated back to Augustine (Augustine | Wittgenstein 1953). Since the 1970s, this school of thought has undergone dramatic revival the quest for symbolized manipulation in cognitive modeling, such wew
as "language of thought" (Fodor) 1975). "physical symbol system" (Newell, 1980), and "algebraic mind" (Marcus 2001). In their study, connectionist's s specific superiority and mability to gener- alize beyond training (Kansky et al. 2017 Chollet 2019 Santoro al. 2018 Zhang et al., 2019a) have been hypothetically linked to a lack of such symbolized algebraic manipulation (Lake & Ba-
roni 2018 Chollet 2019; Marcus 2020). With evidence that an algebraic treatment adopted in early human development (Marcus et al. 1999) can potentially address the issue (Bahdanau et al. 2019
Mao et al.] 2019 Marcus] 2020), classicist (Fodor et al.) 1988) approaches for generalizable reason- ing used in programs (McCarthy 1960) and blocks world (Winggrad 1971) have resurrected. As a hybrid approach to bridge connectionist and classicist, recent developments lead to neuro-symbolic architectures. In partieular, Yi et al. (2018) demonstrate a neuro-symbon prototype for visual ques- tion answering, where a perception module and a language parsing module are separately trained, and the predefined logic operators associated with language tokens are chained to process the visual information, Mao et al. (2019) soften the predefined operators to afford end-to-end training with only question answers. Hann et al. (2019) and Yi et al. (2020) use the hybrid architecture for metaconcept learning and temporal causal learning, respectively. ALANS follows the classicist's 's call but adopts a neuro-semi-symbolit architecture: it is end-to-end trainable as opposed to Yi et al. (2018 2020) and the operator can be learned and concluded on-the-fly without manual specification (Yi et al.
2018 Mao al. 2019 Han al. 2019
Yi al. 2020).
A Astracct Visual Reasoning Recent works by Santoro et al. 2018) and and /hang et al. (2019a) arouse
the community's interest in abstract visual reasoning. where the task of Raven's 's Progressive Matri- ces (RPM) is introduced as such measure for intelligent agents. Initially proposed as an intelligence
quotient test for humans (Raven 1936 Raven & Court, 1998), RPM is believed to be strongly cor- related with human's i general inte ligenee (Carpenter al. (990) and fluid intelligence (Spearman 1923 1927: Hofstadter] 1995 Jaeggi et al.) 2008). Early RPM-solving systems employ symbolic representations based on hand-designed features and assume access to the underlying logics (Car- penter et al.) 1990 Lovett et al., 2009 2010 Lovett& & Forbus, 2017). Another stream of research on RPM recruits similarity-based metrics to select the most similar answer from the choices (Lit- al., 2014: Mekik al.] 2018 Shegheva
tle et al. & Goel 2018). However, their hand- visual features are unable to handle uncertainty from impertect perception, and directly assuming access to the logic operations simplifies the problem. Recently proposed data-driven approaches arise from the availability of large datasets: [Santoro al. 2018) extend pedagogical RPM generation method (Wang & Su. 2015). whereas ZZhang et al. (2019) use a stochastic image grammar (Zhu et al. (207)) and introduce structural annotations in it, Wuich et al. (2020) further refine to avoid shortcut solutions by statistics in candidate panels. De- spite the fact that RPM intrinsically requires one to perform abstraction, algebraization, induction, and generalization, existing methods bypass such cognitive requirements using a single feedforward pass in connectionist models: [aantoro et al.] (2018) use a relational module (Santoro et al. 2017). Steenbrugge al. (2018) augment it with VAE (Kingma & Welling 20013. Zhang et al. 201 assemble a dynamic tree, Hill et al. (0019) arrange the data in a contrasting manner, Zhhang et al. (2019b) propose a contrast module, Zheng et al. (2019) formulate it in student-teacher setting, Wang et al. (2020) build a multiplex graph network, Hu et al. (2020) aggregate features from a hi- erarchical decomposition, and Wu et al. (200) apply a scattering transformation to learn objects, attributes, and relations. In contrast, ALANS attempts to fulfill the cognitive requirements in neuro-semi-see sembiicymoolic framework: the perception frontend abstracts out visual information, and the reasoning backend induces the hidden operator in algebraic structure.
2012; McGreggor & Goel [014 McGreggor
3 THE ALANS LEARNER
In this section, we introduce the ALANS2 learner for the RPM problem. In each RPM instance, an agent is given an incomplete 3 x 3 panel matrix with the last entry missing and asked to induce the operator hidden in the matrix and choose from eight choice panels one that follows it. Formally, let the answer variable be denoted as y,. the context panels as (Lo.33_]. and choice panels as {Je i=1 According to the
Then the problem can be formulated as estimating P2y (o..) (a..) common design (Santoro et al., 2018; Zhang et al.) 2019a Carpenter et al., 1990). there is one operator that governs each panel attribute. Hence, by assuming independence among attributes, we
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An overview of the ALANS* learner. For an RPM instance, the neural visual perception module produces the belief stales for all panels: object CNN extracts object attribute distributions for each image region, and belief inference engine marginalizes them panel attribute, the algebraic abstract reasoning module transforms the belief states into matrix based algebraic representations and induces hidden operators by solving inner optimizations. The answer representations are obtained by executing the induced operators, and the choice most similar to the prediction is selected as the solution. An example of the underlying discrete algehra and its correspondence is also shown the right.
Figure 1:
obtain panel attribute distributions. For each
propose to factorize the probability as Pι ( ))x ITΣ P(L = T", (1,,(,.)) P(T (Ta,))=1),
T
where yd denotes the answer selection based only on attribute and T the operator
Overview As shown in Fig. D the ALANS² learner decomposes the process into perception and reasoning: the neural visual perception frontend extracts the belief states from each of the sixteen panels, whereas the algebraic abstract reasoning backend views instance as example in an abstract algebra structure, transforms belief states into algebraic representations by representation theory, induces the hidden operators, and execuies the operators to predict the representation of the answer. Therefore, Eq. (I). the operator distribution modeled by the fitness of an operator and the answer distribution by the distance between the predicted representation and that of a candidate.
3.1 NEURAL VISUAL PERCEPTION
The neural visual perception frontend consists of an object CNN and a belief inference engine. It is responsible for extracting the belief states for each of the sixteen (context and choice) panels.
Object CNN For each panel, we use a sliding window to traverse the spatial domain of the image and feed each image region into an object CNN. The CNN has four branches, producing for each region its object attribute distributions, including objectiveness (if the region contains an object), type, size, and color. Distributions of type, size, and color are conditioned on an object's existence.
Belief Inference Engine The belief inference engine summarizes the panel attribute distributions (over position, number, type, size, and color) by marginalizing out all object attribute distributions (over objectiveness, type, size, and color). As an example, the distribution of the panel attribute of Number can be computed as such: for N image regions and their predicted objectiveness
P(Number k) Σ P(r RT), R
where P(rg)) denotes the jth region's 's estimated objectiveness distribution, and Ro is binary se- quence of length N that sums k. All panel attribute distributions compose the belief state of panel. In the following, we denote the belief state as and the distribution an attribute a as P(b)).
3.2 ALGEBRAIC ABSTRACT REASONING
Given the belief states of both context and choice panels, the algebraic abstract reasoning backend concerns the induction of hidden operators and the prediction of answer representations for each
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attribute. The fitness of induced operators is used for estimating the operator distribution and the difference between the prediction and the choice panel for estimating the answer distribution.
Algebraic Underpinning Without loss of generality, here we assume row-wise operators. For each attribute, under perfect perception, the first two rows in an RPM instance provide snapshots into an example of magma (Hausmann & Ore] 1937) constrained to an integer-indexed set, the sim- plest group-like algebra structure that is closed under a binary operator. To see this, note that an accurate perception module would see each panel attribute as a deterministic set element. Therefore, RPM instances with unary operators, such as progression, are magma examples with special binary operators where one operand is constant. Instances with binary operators, such arithmetics, di- rectly follow the magma properties. Those with ternary operators are ones with unary operators on a three-tuple set defined on rows.
systematic algebraic view al- lows us to felicitously recruit ideas in representation the- ory (Humphreys 2012) to glean the hidden properties in the abstract structures: it makes abstract algebra amenable by re- ducing it onto linear algebra. Following the same spirit, we pro- pose to lift both the set elements and the hidden operators a learnable matrix space. To encode the set element, we em- play the Peano Axiom (Peano 1889). According to the Peano Axiom, an integer-indexed set can be constructed by (1) zero element (0), (2) a successor function (S(-)), and (3) the prin- ciple of mathematical induction, such that the kth element encoded S (0). Specifically,
Algebraic Representation
(( and srrred
Figure 2: Isomorphism between the abstract algebra and the matrix-based representation. Operator induction re- duced to matrices.
instantiate the successor function as the matrix-matrix product parameterized by M. In an attribute-specific man- ner, the representation of an attribute taking the kth value is (Ma)* M8- For operators, we consider them to live in a learnable matrix group of a corresponding dimension, such that the action of an op- erator set can be represented as matrix multiplication. Such algebraic representations establish an isomorphism between the matrix space and the abstract algebraic structure: abstract elements on the algebraic structure have bijective mapping to/from the matrix space, and inducing the abstract relation can be reduced to solving for a matrix operator. See Fig. 2 for a graphical illustration of the isomorphism.
element as a learnable matrix Mo and the
Operator Induction Operator induction concerns about finding a concrete operator in the abstract algebraic structure. By the property of closure, we formulate as an inner-level regularized linear regression problem: a binary operator T, in a magma example for attribute a minimizes arg min (T) = Σ [].M(CE))TMMEEEC
M( + AgTTE
(3)
where under visual uncertainty, we take the expectation with respect to the distributions in the belief states of context panels P(bS.) in the first two rows, and denote its algebraic representation as M (bg.). ). For unary operators, one operand can be treated as constant and absorbed into T. Note that Eq. 33) admits a closed-form solution (see Appendix for details). Therefore, the operator can be learned and adapted for different instances of binary relations and concluded on- on-the-fly. Such a design also simplifies the recent neuro- symbolic approaches. where every single symbol operator needs to be hand-defined (Yi et al. 2018 Mao al. 2019 Han et al. 2019 Yi et al. 2020). Instead, we only specify an inner-level optimization framework and allow symbolic operators to be quickly induced based on the neural observations, while keeping the semantic interpretability in the neuro- symbolic methods. Therefore, we term such a design semi-symbolic.
The operator probability in Eq. I is then modeled by each operator type's fitness, e.g., for binary, P(T TT (T,)) xc exp(-e;(TT))
(4)
Operator Execution To predict the algebraic representation of the answer, we solve another inner- level optimization similar to Eq. (3). but now treating the representation of the answer as a variable:
Mg arg min (M) E[JM TT, M(E Ml}-)
(5)
where the expectation is taken with respect to context panels in the last row. The optimization also admits closed-form solution (see Appendix for details), which corresponds to the execution of the induced operator in Eq.
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The predicted representation is decoded probabilistically as the predicted belief state of the solution,
p(tr T x exp(-IIM* - (M*)* M5 (?2)
(6)
Answer Selection Based on Eqs. ① and 4. estimating the answer distribution is now boiled down estimating the conditional answer distributions each attribute. Here, we propose model it based on the Jensen-Shannon Divergence (ISD) of the predicted belief state and that of a choice,
P(y" n T", (lo,))=1 (( x exp(-DDsD(P(C)) TT))P((EE..)))
(7)
Discussion The algebraic abstract reasoning module offers a computational and interpretable counterpart human-like reasoning in RPM (Carpenter et al.) 1990). Specifically, the induction component resembles the fluid intelligence, where one quickly induces the hidden operator by ob- serving the context panels. The execution component synthesizes operator, and the choice most similar to the image is selected as the answer. We also note that by decoding the predicted representation in Eq. (6), a solution can be generated: by sequentially se- lecting the most probable operator and the most probable attribute value, a rendering engine can directly render the solution. The reasoning backend also enables end-to-end training: by integrating the belief states from neural perception, the module conducts both induction and execution in a soft manner, such that the gradients can be back proppaatt and the learner jointly trained.
image by executing the induced
3.3 LEARNING OBJECTIVE
We train the entire ALANS" learner by minimizing the cross-entropy loss between the estimated answer distribution and the ground-truth selection, i.e.,
min ((P(y)
(e,. y.).
6,4Mg )-
where E(-) denotes the cross-entropy loss, y. the ground-truth selection, 9 the parameters in the
object CNN, and { [Mā} and { (Ma) the zero elements and the successor functions for element encod- ings, respectively. Note notations are simplified by making the dependency on parameters implicit.
However, we notice in practice that with only the cross-entropy loss on the ground-truth selection, the ALANS² learner experiences difficulty in convergence. Without a proper guidance, the object CNN does not produce meaningful object-based representations, Therefore, following the discussion in (Santoro et al., 2018; Zhang et al., 2019a; Wang et al., 20200) we augment training with an auxiliary loss on the distribution of the operator, i.e..
min 6.4Mg (
((P(g) (To.l 1......).... + Σ XX(((TTT (To.))
(9)
where y. denotes the ground-truth operator selection for attribute and x balances the trade-off.
4 EXPERIMENTS
cognitive architecture with systematic generalization is believed to demonstrate the following three principles (Fodor et al., 1988) Marcus 2001] 2020): (1) systematicity, (2) productivity, and (3) localism. Systematicityy requires an architecture to be able entertain "semantically related" contents after understanding a given thought. Productivity states that the awareness of a constituent implies that of recursive application of the constituent, and vice versa for localism.
To verify the effectiveness of an algebraic treatment in systematic generalization, we showcase the superiority of the proposed ALANS learner on the three principles in the abstract spatial-temporal reasoning task of RPM. Specifically,
use the generation methods proposed in Zh/Zhng et al. (2019a) and et al. (2020) to generate RPM problems and carefully split training and testing to construct the three regimes. The former generates candidates by perturbing only one attribute of the correct answer while the later modifies attribute values in a hierarchical manner avoid shorteut solutions by pure statistics. Both methods categorize relations in RPM into three types, according to Carpenter et al. (1990): unary (Constant and Progression), binary (Arithmetic), and ternary (Distribution of Three), each of which comes with several instances. Grounding the principles into learning abstract relations in RPM, we fix the configuration to be 3 x 3Grid and generate the following data splits for evaluation (see Appendix for details):
Systematicity; the training set contains only subset of instances for each type of relation, while the test set all other relation instances.

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Productivity: as the binary relation results from recursive application of the unary relation, the training set contains only unary relations, whereas the test set only binary relations. Localism: the training and testing sets in the productivity split are swapped to study localism.
We follow Zhang al. (2019a) to generate 10, 000 instances for each split and assign 6 folds for training, 2 folds for validation, and 2 folds for testing.
Experimental Setup We evaluate the systematic generalizability of the proposed ALANS learner on the above three splits, and compare the ALANS" learner with other baselines, including ResNet, ResNet+DRT (Zhang et al. 22019)), WReN (Santoro et al., 2018), CoPINet Zhhag et al. 2019b).
MXGNet (Wang al. 2020)) LEN (Zheng et al., 2019), HriNet (Hu et al. 2020), and SCL (Wu ct al. 2020). We use either official or public implementations that reproduce the original results.
Table I: Model performance on different aspects of systematic generalization. The performance is measured by accuracy and reported on the test sets, Upper: results on datasets generated by Zhang et al. (2019a) Lower: results on datasets generated by Hu et al. (2020).
Method
MXGNet
ResNe++DRT
ResNet
HriNet
LEN
WReN
SCL.
CoPINet
ALANS
ALANS² Ind
Systematicity
20.95%
33.00%
27.35%
28.06%
40.15%
35.20%
37.35%
59.30%
78.45%
52.70%
Productivity
30. 41%6
27.93%
27.05%
31.45%
12.30%
56.95%
51.10%
60.001%
79.95%
36.15%
Localism
28. 80%
21.90%
23.05%
29.70%
39.65%
38.70%
47.75%
60.10%
80. 80.50%
59.80%
Average
26.72%6
28.62%
25.82%
29.73%
40.70%
43.62%
45.40%
59.80%
79.63%
48.65%
Systematicity
13.35%
13.50%
14.20%
21.00%
17.40%
15.00%
24.90%
18.35%
64. 64.80%
52.80%
Productivity
14.10%
16.10%
20.70%
20.35%
19.70%
17.95%
22.20%
29.10%
65.55%
32.10%
Localism
15.8096
13.83%
17.45%
24.60%
19.70%
99..95%
31. 85%
65. 90%%
50.70%
Average
14.42%
14.48%
17.45%
21.98%
19.08%
17.56%
25.68%
26%4
65.42%
45.20%
Systematic Generalization Table D shows the performance of various models on systematic gen- cralization, i.e., systematicity, productivity, and localism. Compared to results reported in Santoro al.] 20188) Zhang al. (2019a b): Wang et al. (2020); Zheng et al. (2019); Hu al. (020)] Wu et al. (220). all pure connectionist models experience a devastating performance drop when it comes to the critical cognitive requirements systematic generalization, indicating that pure connection- ist models fail to perform abstraction, algebraization, induction, or generalization needed in solving the abstract reasoning task; instead, they seem to only take a shortcut to bypass them. In particular, MXGNet (Wang et al., 2020)'s 's superiority is diminishing in systematic generalization. Despite of learning with structural annotations, ResNet+DRT (Zhang et al., 2019a) does not fare better than its base model. The recently proposed HriNet (Hu et al.) 2020) slightly improves on ResNet in this aspect, with LEN (Zheng et al.) (219)) being only marginally better. WReN (Santoro et al.. 2018), on the other hand, shows oscillating performance across the three regimes. Evaluated under systematic generation, SCL (Wu al. 2020) and CoPINet (Zhang et al. 2019b) also far deviate from their "su- perior performance" These observations suggest that pure connectionist models highly likely learn from variation in visual appearance rather than the algebra underlying the problem.
Embedded in a soural-semi-symboll framework, the proposed ALANS learner improves on sys- tematic generalization by a large margin. With an algebra-aware design, the model is considerably stable across different principles of systematic generalization. The algebraic representations learned in relations of either a constituent or a recursive composition naturally support productivity and lo- calism, while semi- semi-symbolic inner optimization further allows various instances of an operator type to be induced from the algebraic representations and boosts systematicity. The importance of the al- gebraic representations is made more significant in the ablation study: ALANS" Ind, with algebraic representation replaced by independent encodings and the algebraic isomorphism broken, shows inferior performance. The ALANS² learner also enables diagnostic tests into its jointly learned per- ception module and reasoning module, in contrast to the black-box-like connectionist counterparts.
Analysis into Perception and Reasoning The neural-semi-symbolic design allows analyses into both perception and reasoning. To evaluate the neural perception module and the algebraic reason- ing module, we extract region-based object attribute annotations from the dataset generation meth- ods (Zhang et al., 2019a Hu et al. 2020) and categorize all relations into three types, i.e., unary, binary, and ternary, respectively.
shows the perception module's performance on the test sets in the three regimes of systematic generalization. We note that in order for the ALANS² learner to achieve the desired results shown in Table [] ALANS learns to construct the concept of objectiveness perfectly. The model also shows fairly accurate prediction accuracy on the attributes of type and size. However, on the texture- concept of color, ALANS² fails to develop a reliable notion on it. Despite that, the general prediction accuracy of the perception module is still surprising, considering that the perception module is only
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jointly learned with ground-truti annotations answer selections. The relatively lower accuracy color could be attributed to its larger space compared to other attributes.
Table 2: Perception accuracy of the proposed ALANS learner, measured by whether the modulc can correctly
predict an attribute's s value. Left: results on datasets genereted by Zhang et al. (2019a). Right: results on datasets
genereted by Hu al. (2020)
Object Atribute
Objectiveness
Type
Size
Color
Object Attribute
Objectiveness
Type
Size
Color
Systematicity
100.00%
99.95%
94.65%
71.35%
Systematicity
100.00%
96.344%
92.36%
63.98%
Productivity
100.00%
99.97%
98. 04%
77.61%
Productivvty
100.00%
94.28%
97.00%
69.89%
Localism
100.00%
95.63%
98.56%
80.05%
Localism
100.00%
95.80%
98.36%
80.35%
Average
100.00%
98.52%
97.08%
76.34%
Average
100.00%
95.47%
95.91%
64.74%
Table 3: Reasoning accuracy of the proposed ALANS learner, measured by whether the module can correctly
predict the type of relation an attribute. Left: results datasets genereted by Zhang al. (2019a). Right: results on datasets generated by Hu et al. (2020).
Relation on
Position
Number
Type
Size
Color
Relation
Position
Number
Type
Size
Color
Systemaricity Productivity
72.04%
82.14%
81.50%
80.80%
40.40%
Systematicity
69.96%
80.345
83.50%
80.85%
28.85%
98.75%
89.501
72.10%
33.95%
Productivity
90.10%
87 95%9%
68. 50%
23. 10%
Localism Average
74.70%
44.25%
10%
51.20%
Localism
70.55%
36 36.65%
42.30%
33.20%
72.04%
85.20%
71.73%
69.77%
42.85%
Averuge
69.96%
83.33%
69.37%
63.88%
28.38%
Table 3 lists the reasoning module's performance during testing for the three aspects. Note that on position, the unary operator (shifting) and binary operator (set arithemtics) do not systematically im- ply each other. Hence. we do not count them as probes into productivity and localism. general, we notice that the hetter the perception accuracy on one attribute, the better the performance on reason- ing. However, we also note that despite the relatively accurate perception of objectiveness, type, and size, near perfect reasoning never guaranteed. This deficiency due to the perception uncertainty handled by expectation in Eq. (3: in spite of correctness when we take arg max, marginalizing by expectation will unavoidably introduce noise into the reasoning process. Therefore, ideal reason- ing module requires the perception frontend to be not only correct but also certain. Computationally. one can sample from the perception module and optimize Eq. (9) using REINFORCE (Williams) 1992). However, the credit assignment problem and variance gradient estimation will further complicate training.
Generative Potential Compared existing discriminative-only RPM-solving methods, the pro- posed ALANS learner is unique in its generative potential. As mentioned above, the final panel attribute can be decoded by sequentially selecting the most probable hidden operator and the at- tribute value. When equipped with a rendering engine, solution can be generated. Here, we use the rendering program released by Zhang et al. (2019a) to demonstrate such a generative potential in the proposed ALANS* learner. Fig. 3 shows examples where the solutions are generated by ALANS". Such a generative ability is computational counterpart to human reasoning: ALANS selects the one most similar to synthesized image from the pool of candidates, which resembles human's top-down bottom-up reasoning.
Position
N/A
Poeition
Distribute Three
Number
Progrossion
Namber
:
N/A Distribute Throe
Type
Distribbuee Three
Type
Sizee Color
Distriiutt Three
Size Color
Progression
Distribute Three
Prograssion
Figure 3: Examples of RPM instances with the missing entries filled by solutions generated by the *LANS learner. Ground-truth relations are also listed. Note the generated results do look exactly like the correct choices due to random rotations during rendering. but they are semantically correct.
5 CONCLUSION
In this work, we propose the Algebra-Awart Neuro-Semi-Se SemiiSSmbollic (ALANS?) learner, echoing a normative theory in the connectionist-clasiii debate that algebraic treatment in cognitive ar- chitecture should improve model's systematic generalization ability, In experiments, we show that with such an algebraic treatment, the neuro-semi-symboul learner achieves superior performance in three RPM domains reflective of systematic generalization.
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A INDUCING AND EXECUTING OPERATORS
In the main text, we examplify the induction and the execution process using a binary operator Here, we discuss other details regarding the formulation for all three types of operators, i.e., unary, binary, and ternary.
Unary Operator mization problem
To induce the unary operator T for an attribute a, we solve the following opti-
The = arg min (7) = 1/5 x
E | (6
+ E [ ((,,,))) F
- -
>)l
+ E [M M(
[I M(
x 1
(S1)
where the indexing follows the row column major. By taking the derivative with respect to T and setting to be 0, we have the following solution,
T A
(S2)
where, assuming independence,
A =E (bS,,) M( E M ) M 2) E (M(C,)) M(BS))
(S3)
[M M(BS)
[M M(C [7) 5
and
B E [M ""
M (6 [2 +
E [M
E M (b0,3) + E [M J [M(s,))
E (b )' E M (, + (M E ((b5s)))
(S4)
Note that as long as x > 0, is a symmetric positive definite matrix and hence is invertible. Compared to the binary case, the unary operator can be regarded as special binary operator where one of the operand is a constant, absorbed into operator learning, and jointly solved.
To predict the answer representation, we solve another optimization problem, i.e.
Mg arg min (M) E M( M | M
(S5)
Taking its derivative and setting it to 0, we have
Mg E ( T.
(S6)
Note that this is exactly the execution of the learned operator. Binary Operator
The optimization problem for the binary case can be expanded To arg min (8( 1/2
I (TM0)))) M(88
T
(S7)
E
M( M(GD)) Ag 1T?
We note that, assuming independence, the solution satisfies
E [ 1 M TIE 2 M(b., ["] '1++
E [M MM A TE [M ( M [ 2ALT
(S8)
E [<(((...) E [MC,C.)) E [M()) E [MC ( (E [M [M
=
This a linear matrix equation and can be turned into a linear equation by vectorization. Using
vec(ATB) A B vec(T) (Lancaster 1970). where @ denotes the Kronecker product, we have
vec(7,)) BB
(S9)
where
A E (M(SE)
X E M(C
(S10)
E 1f
E
(MC + 2XgT
M
and
B vec (E [ (b. 1 M (
[M( +
(S11)
vec (E [M ) E [ (bC.,) E [ba,s)
Note that A is also symmetric positive definite given positive Ag and hence invertible.

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The predicted answer representation is given by
Mg arg min cg (M) E (1 (b -M
(S12)
M
which can be solved by executing the induced binary operator Mg. = E [M( T (b,,)
Ternary Operator A ternary operation can be regarded as unary operation elements defined
on rows / columns. Specifically, we propose to construct the algebraic representation of row / column by concatenating the algehraic representation of each panel in it, Le.,
M(bg, 16,++1, 1o (2,1+2)
[M (b, (Bb,,++)
(S13)
Then the ternary operator can be solved by
7,e = arg min (T) = || 2 b M(b 8y,5 8g |
+ 22 IT
(S14)
Similar to the unary case discussed above,
T = AB
(S15)
where
A = E [M 2, k2,)²² ( MM, by b%,3) A
(S16)
and
== = B [M 62 b8,)) (6,, &2,5, (%,6)].
(S17)
Correspondingly, the answer representation can be obtained by first executing the ternary operator E [b 02,5, 82 0%,6)] [ T," and slicing it from the result.
To compute the operator distribution, we model based on the fitness of each operator type,
P(T Tu (To.) cc exp(-f"
(S18)
P(T Th"
(4o,ih 8 o exp(-(g(76.)
(S19)
P(T T," (To.) 0c exp( (T
(S20)
B INSTANCES OF OPERATORS
In the original work of Zhang et al. (2019a) and Hu et al. 2020)) there four operators: Con- stant, Progression, Arithmetic, and Distribute of Three, Progression is parameterized by its step size (+1/2). Arithmetic includes addition and subtraction. And Distribute of Three is implemented as shifting and can be either a left shift or right one. Note that Constant can be regarded as special Progression with a step size of O. In this work, group all four operators into three types: unary (Constant and Progression), binary (Arithmetic), and ternary (Distribute of Three).
To study systematic generalization in abstract relation learning, proposed in (Zhang et al. 2019a Hu et al. 2020) and carefully split data into three regimes:
the RPM generation method
The training set and the test set contain all three types of operators but disjoint instances. Specifically, the training set has Constant, Progression of 1, addition in Arithmetic, and left shift in Distribute of Three, while in the test set there are Progression of +2 subtraction Arithmetic, and right shift in Distribute of Three
Systematicity:
Productivity:
The training set contains only unary operators and the test set only binary operators. Specifically, the training set has Constant and all instances of Progression, while the test set all instances of Arithmetic.
Localism: The training set contains only binary operators and the test set only unary operators. Specificallyy the training set has all instances of Arithmetic and the test set Constant and all in- stances of Progression.
Please see Figs. to for examples in the three splits.
C IMPLEMENTATION DETAILS
C.I NETWORK ARCHITECTURE
We use a LeNet-like architecture (LeCun et al. 1998) for each branch of the object CNN. See for the design. Note that the object CNN consists of four branches, including objectiveness,
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A
Figure S1: A training example (left) and test example (right) in the systematicity split. Note that in the training example, the arithmetic relation (in number) addition and the shifting is always a left shift (in type, size, and color). In test example, the shifting becomes right shift (in type), the size progression has step and color arithmetic becomes subtraction.
?
Figure training example (left) and a test example (right) in the productivity split. Note that the training example, the constant rule applied to the number, type, and size, while the progression rule is applied on color. In the testing example, the arithmetic rule is applied on all attributes.
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4
4
4
A
A
?
A
Figure S3: training example (left) and test example (right) in the localism split. Note that in the training example, the arithmetic rule is on all attributes. In the test example, the progression rule is applied on number and the constant rule all other attributes.
Table S1: The network architecture used for each branch of the object CNN.
Operator
Parameters
Convolution
[6. 5, 1]
BatchNorm
6
SoftPlus
MaxPool
2
Convolution
[16,5, 1] 16
BatchNorm
SoftPlus
MaxPool
2
Linear
120
SoftPlus
Linear
84
SoftPlus
Linear
LogSoftMax
type, size, and color. The parameters for Convolution denote the output channel size, kernel size, and stride, respectively. A BatchNorm layer is parameterized by the number of channels, whereas MaxPool layer by its stride. An output size used specify Linear layer's parameter. m equals 2, 5. 6, 10 for objectiveness. type, size, and color, respectively. For numerical stability, we use LogSoftMax to turn probability simplex into its log space.
C.2 OTHER HYPERPARAMETERS
For the inner regularized linear regression, we set different regularization coefficients for different attributes but, for the same attribute, we keep them the same across all three types of operators. For position, À 10-1 For number, À 10-6 For type, x 10-6 For size, À 10-6 For color, 10-7, All of the regularization terms Eq. (9) in the main text are set be and (M8) and (M*) initialized 2 x 2 square matrices.
For training, we first train for 10 epochs parameters regarding objectiveness, including the objective- ness branch, and the representation matrices on position and number. We then perform 2 rounds of cyclic training on parameters regarding type, size, and color, each of which experiences 10 epochs of updates in a round. Finally, fine-tune all parameters for another 10 epochs, totaling up to 80 training epochs. The entire system is optimized using ADAM (Kingma & Ba 2014) with a learning rate of 9.5 50
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D MARGINALIZATION FOR OTHER ATTRIBUTES
For the attribute of position, we denote its value as R a binary vector of length N, with each entry corresponding to one of the N windows. Then
PiPosition [2') II PGr R2))
(S21)
where P(r3) denotes the jth region's 's estimated objectiveness distribution returned by a CNN as in the main text.
For the attribute of type, the panel attribute of type being le is evaluated as
P(Type k)) Σ
11
P(r) -k) P(Position R²);
(S22)
(.R5=1
where P(r)) denotes the jth region's 's estimated type distribution returned by CNN.
The computation for size and color is exactly the same as type, except that we use the region's estimated size and color distribution returned by a CNN.
E RELATED WORK ON NEURAL THEOREM PROVING
Combining neural architectures with symbolic reasoning has a long history in the field of theorem proving Garcez et al., 2012). with early works dated back to propositional rules (Shavlik & Towell
1991 Towell & Shavlikk 1994; Garcez& Zaverucha 1999). Later works extend the propositional
rules to first-order inference (Shastri 1992 Ding 1995 França et al.] 2014; Sourek et al. [015 Co-
hen 2016). More recent works include the Logic Tensor Networks (Serafini Garcez, 2016 6) and
NTP model (Rocktäschel & Riedel, 2017). The former grounds first-order logics and supports func-
tion terms, while the latter is constructed from Prolog's backward chaining and is related to Komen- dantskaya 2011); Hiolldobler (1990) but supports function- -free terms. DeepProbLog (Manhaeve et al. 2018) further improves on NTP by focusing on tight interactions between a neural component and subsymbolic representation and parameter learning for both the neural and the logic compo- nents. [vans & Grefenstette (2018) introduces a differentiable rule induction process, though not integrating the neural and symbolic components. Our work is related to the stream of work on neural theorem proving. However, we formulate the relation induction process as continuous optimization rather than logical induction.
F
MORE ON NEURAL VISUAL PERCEPTION
Why not train CNN to predict the position and number of objects? The CNN is trained to predict the type, size, color, and object existence window. The object ex- istence in windows is marginalized to be Number distribution and Position distribution. This is a light method for object detection. Nevertheless, it is also possible to use Fast-RCNN like method to predict object positions (this implies number) directly. How- ever, in this way, the framework loses the probabilistic interpretation (the object proposal branch is currently still deterministic), and cannot perform end-to-end learning.
How does the CNN predict the presence of object, its type, size, and color given that it is not trained to do that? For each window, the CNN outputs softmaxed vec- tors, corresponding the probability distributions of object existence, object type, object size, and object color. The spaces for these attributes are pre-defined. CNN's weights are then jointly trained in the framework. Such design follows recent neuro-symbolic meth- ods (Mao et al. 2019: Han et al., 2019) that also rely on the implicitly trained represen- tation. In short, we assign semantics the implicitly trained representation (probability distributions for attributes), performs marginalization and reasoning as if they are ground- truth attribute distributions, and jointly train using only the problem's target label.
5


Reviews:
[{"review_id": "wJnh5H9k8Z", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "This paper proposed ALANS, a semi neuro-symbolic learner specifically designed for improved systematic generalization on RAVEN dataset.\n\nPros:\n1. The proposed model integrates learnable symbolic operators that are similar to the ones used in Neural Theorem Proving. The operators are shown to have improved systematic generalization performance.\n2. Introduced a new generalization testing dataset based on RAVEN dataset.\n\nCons:\n1. The proposed model relies on strong assumptions specific to RPM, and even RAVEN dataset. For example, In Equation 3, the authors assume the knowledge that relations can only exist in rows of the RAVEN dataset. However for PGM dataset (another well-known RPM dataset), the relations can exist either in rows or columns, or both. I hypothesize that training operators when there is no guaranteed existence of relations will lead to noisy gradients. I have doubts that the proposed method can be readily adapted to PGM, not to mention other types of reasoning tasks.\n2. While the proposed model is shown to perform well on the dataset generated based on RAVEN, I think the authors should show results on PGM dataset as well, considering that PGM already have a few data splits to test systematic generalization.\n\nClarity:\nIn general the paper is written in a clear and understandable manner. The only part that confuses me is about the perception CNN module. It is not clear to me how the model learns to disentangle object attributes such as color and position? Is there any auxiliary labels to help induce such disentanglement, or does the model learn to disentangle the attributes because of the way authors marginalize the object attributes in a diagram? The authors only show the marginalization for the attribute 'number'. I suggest the authors to include the marginalization for all other attributes in the Appendix for better understandability. \n\nSummary:\nOverall I think this paper propose an interesting approach that improves systematic generalization on RAVEN datasets. But I have doubts on its adaptabilities to PGM dataset and other types of reasoning tasks. And there is also some lacking details that hinders understanding of the paper. I will raise my score if the authors can (1) show improved performance on generalization data splits on PGM dataset, and (2) clarify about how the model learns to disentanglement the object attributes.\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "Z8GIw34AoSu", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "**Summary**.\nThis work proposes a new learner bridging the gap between connectionists and classicists in the task of Raven’s Progressive Matrices (RPM). It relies on a CNN to extract visual features and then uses an algebraic abstract reasoning module to infer the operators of an RPM instance, which allows applying the inferred operator on the RPM instance to predict potential solutions according to various attributes. The most likely solution according to the ensemble of the attributes is then selected as an answer.\n\n**Pros**.\n- This work is significant as neural models still struggle in systematically generalizing in reasoning tasks.\n- The general idea of the paper is easy to follow but that does not mean the proposed method is trivial, far from it.\n- Experiments are clearly described and the authors give particular attention to split their dataset to systematically test for generalization.\n\n**Cons**.\n- Additional clarifications and motivations could benefit the paper (see **Questions & Suggestions** below).\n- While yielding impressive results and targeting a fundamental issue of neural models, it is not clear how the proposed technique could be applied to other domains. As of now the paper focus only on the RPM task. It would be beneficial to open up the discussion to potential applications. It would be nice to know what the authors think are the potential applications of their learner. How would it be applied to other reasoning tasks?\n\n**Questions & Suggestions**.\n- The CNN is used to predict the presence of an object, its type, its size, and its color while a belief inference engine is used to predict the position and number of objects. It is not clear why the CNN was not also used to predict the position and number of objects, thus questioning the motivation behind the belief inference engine.\nWould training a CNN to predict the position and number of objects also help?\n- It is not clear how the CNN predicts the presence of an object, its type, size, and color given that it is not trained to do that. Do you produce 4 feature vectors, apply a softmax to each, and arbitrarily decide which feature vector will predict each object attribute? If so the results in Table2 are very impressive and hard to believe. How do you decide which feature vector will predict each object attribute?\n- Suggestion: as an ablation study, it would be nice to also compare the performance of ALANSS with randomly initialized and fixed CNNs. This would give an estimate on the advantage of using CNNs as neural modules. One could hypothesize that the robustness of the method comes mostly from the symbolic abstraction rather than the neural representation.\n- Neural Theorem Proving also tries to bridge the gap between connectionists and classicist approaches. Can it be related to this work? It would be nice to discuss this in the Related Work section.\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "XJq4_kMFABe", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "Initial review (2020.11.01)\n\nReview: \nThis paper addresses a specialized solution to combine neural networks and linear models for Raven progressive matrices (RPM). The solver is composed of convolutional neural networks (CNN) as an image perception module and a reasoning module with linearly parameterized operators to represent successive operations in the RPM. The operators are trained with regularized linear regressors, which are lightweight capable to adapt on-the-fly for each instance. The overall architecture enables end-to-end training and can be used for generation. For performance evaluation on two kinds of automatically generated problem sets, the authors test the proposed method with respect to 3 extrapolatory settings such as systematicity, productivity, and localism, which seems to be newly defined tasks for RPM. Their method outperforms pure neural state-of-the-art methods with a large margin over all of settings. \nThe proposed methods solve RPMs well with available components such as CNNs and linear models, which are able to be tuned on-the-fly. Also, this paper provides systematic results to easily comprehend the role of modules with new settings (systematicity, productivity, and localism) for RPMs. On the other hands, my major concerns are two-fold: (1) one of important research questions introduced in this work, “what constitutes such an algebraic inductive bias?” seems not so clearly answered in this paper. (2) the comparative result in the paper just considers only pure connectionist methods without considering other (semi-)symbolic approaches. Since this work utilizes additional problem information (e.g., specified features) for modeling than compared neural methods, I think it would be better to justify the position of this work by comparing various approaches including search-based solver, symbolic, and the neural methods to figure out the superiority of this approach.\nAs a result, I vote for marginally below acceptance threshold.\n\nPros:\n-\tThe authors provide a specialized solution for RPM well with CNNs and linear models.\n-\tThey introduce new 3 settings of RPMs and report systematic experimental results.\n\nConcerns:\n-\tIn the comparative study, the authors report only the systematicity, the productivity, and the localism, not the \"accuracy\" used in the cited related works. Due to the lack of the information, the readers can not directly compare them with respect to the previous viewpoint.\n-\tIf the result of other various approaches for RPM not only pure connectionist methods but also search-based solver, symbolic, and neuro-symbolic approaches, it would be easier to find the niche of this work and its superiority.\n-\tAs I mentioned above, the research questions, “what constitutes such an algebraic inductive bias?” seems not so clearly answered in this paper. Could you discuss this issue with the result in the paper?\n\nMinors:\n-\tIt would be better understandable to show examples for 3 settings of systematicity, productivity and localism on RPMs.\n-\tfare -> far?\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "TH7PBUGUOmu", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 7, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "The author(s) propose an architecture ALANS^2, primarily focused on the task\nof Raven's Progressive Matrices (RPM, popularly known as IQ tests).\nTo solve it, they join an image classifier on the individual fields\nof the 3x3 picture matrix with a \"reasoning backend\". In the backend,\nthey model the high level rules of the task by matrix multiplication.\nGiven a problem instance, they find a such an operator (matrix) that fits\nthe first two rows of the instance the best, and use this operator\nto determine what picture belongs to the missing field.\n\nFor training, they use not only the knowledge of the correct answer but also\nthe correct operator which should be found (the \"rule\" which should be discovered).\nExperimental results show that they exceed by a margin\nthe previous models used for this task.\n\nThe task is interesting, in fact some research has been done on trying to\nlearn intelligence tests in various ways and a number of the relevant ones\nare properly cited in the paper.\n\nThe results presented show an improvement with respect to previous\nresults that I find enough to deserve publication.\n\nThe article is well written apart from the typo mentioned below.\n\nTypo:\nPg3, part \"Abstract Visual Reasoning\" last sentence: fronend -> frontend\n", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper was reviewed by 4 experts in the field. The reviewers raised their concerns on lack of novelty, unconvincing experiment, and the presentation of this paper, While the paper clearly has merit, the decision is not to recommend acceptance. The authors are encouraged to consider the reviewers' comments when revising the paper for submission elsewhere.
Decision: Reject