CoFrNets: Interpretable Neural Architecture Inspired by Continued Fractions
Authors: Isha Puri
Amit Dhurandhar
Tejaswini Pedapati
Karthikeyan Shanmugam
Dennis Wei
Kush R. Varshney
Venue: NeurIPS 2021 Poster (2021)
Keywords: neural architecture design
explainable AI
continued fractions

Abstract: In recent years there has been a considerable amount of research on local post hoc explanations for neural networks. However, work on building interpretable neural architectures has been relatively sparse. In this paper, we present a novel neural architecture, CoFrNet, inspired by the form of continued fractions which are known to have many attractive properties in number theory, such as fast convergence of approximations to real numbers. We show that CoFrNets can be efficiently trained as well as interpreted leveraging their particular functional form. Moreover, we prove that such architectures are universal approximators based on a proof strategy that is different than the typical strategy used to prove universal approximation results for neural networks based on infinite width (or depth), which is likely to be of independent interest. We experiment on nonlinear synthetic functions and are able to accurately model as well as estimate feature attributions and even higher order terms in some cases, which is a testament to the representational power as well as interpretability of such architectures. To further showcase the power of CoFrNets, we experiment on seven real datasets spanning tabular, text and image modalities, and show that they are either comparable or significantly better than other interpretable models and multilayer perceptrons, sometimes approaching the accuracies of state-of-the-art models.

CoFrNets: Interpretable Neural Architecture Inspired by Continued Fractions
Isha Puri'
Amit Dhurandhar
Tejaswini Pedapati IBM Research
Harvard University
IBM Research adhuranQus. i bmm. com
ishapuri@college harvard.
tejaswiniphus. imm. com
Karthikeyan Shanmugam
Dennis Wei
Kush R. Varshney
IBM Research
IBM Research dwei fus..us... ibm. com
IBM Research
karthikeyan, sta.rrrgann..aaaaaaaammaaiii com
krvarshn@us ibm. com
Abstract
In recent years there has been considerable amount research on local post hoc explanations for neural networks. However, work building interpretable neural architectures has been relatively sparse. In this paper, we present a novel neural architecture, CoFrNet, inspired by the form of continued fractions which known to have many attractive properties in number theory, such as fast convergence of approximations real numbers. We show that CoFrNets be efficiently trained as well as interpreted leveraging their particular functional form. Moreover, we prove that such architcetures are universal approximators based on a proof strategy that is different than the typical strategy used to prove universal approximation results for neural networks based infinite width (or depth), which likely to be of independent interest. We experiment on nonlinear synthetic functions and are able accurately model well estimate feature attributions and even higher order terms in some cases, which is testament to the representational power well as interpretability of such architectures. To further showcase the power CoFrNets, we experiment seven real datasets spanning tabular, text and image modalities, and show that they are either comparable or significantly better than other interpretable models and multilayer perceptrons, sometimes approaching the accuracics state-of- totte-of-thcaatt art models.
Introduction
"It is simple. The minute I heard the problem, knew that the answer was continued fraction. Which continued fraction, asked myself. Then the answer came to my mind."
This the response of the mathematics genius Ramanujan to Mahalanobis, who was astounded how he was able to solve the difficult Strand puzzle [30] almost instantancouuly. Besides showcasing the genius of Ramanujan, the puzzle also showcases the power of continued fractions. Continued fractions (CFs), typically represented as a sequence that looks like ladder: do
can represent any real number and any analytic function, including trignometric functions, polynomials, the exponential function, power functions, and special functions like the gamma, hy hyergeooettric, and Bessel functions [8]. To represent arbitrary real numbers, it is sufficient for the ags to be non-negative integers and for the by Analytic functions, which can be represented power series, can also
Work done part of intersship at IBM Research.
35th Conference on Neural Information Processing Systems (NeurIPS 2021).

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be represented in this form. Moreover, CFs are the best rational approximations to a number/function in certain sense [31]. Rational approximations are obtained by curtailing the fraction just before the "+" sign. For instance, in the example CF above ag. Bvaytb and aacraztudbztozd are three 21 different rational approximations. Additional properties of CFs are discussed Section 3.
Given the desirable properties of CFs and noticing their ladder-like structure, we propose a neural architecture inspired by CFs illustrated in Figure 1. In place of the ags, linear functions of the input x E IRP are computed by taking the inner product x: with weight vector wi E IRP in each layer k (or step of the ladder). The reciprocal of the function thus far is applied as nonlinearity in each layer. We refer to this Continued Fraction-inspired neural network as CoFrNet (pronounced "coffer net'). (Like coffers in building architecture [49], the proposed neural architecture is made up of repeating structures.) Although more complicated functions than linear could be used in each layer, we show in Section 5 that linear functions are sufficient for universal approximation with a finite number of such ladders. The proof follows a different strategy than typical results on universal approximation of neural networks that rely on the results of Cybenko [9], Hornik [21]. and Zhou [27], and may be of independent interest.
The proposed architecture is "simple": there are only p weights to be learned in each layer as opposed quadratic number for standard ar- chitectures such as multilayer perceptron (MLP) or those with densely connected layers.² A key differentiation from other neural architec- tures that the input is passed into every layer [15, 19, 16, 47].* Moreover, the nonlincarity 1 different from more commonly used nonlin- earities such as sigmoid, ReLU, and polynomi- als. The CF representation is much more com- pact than directly representing polynomial of the same degree, which requires exponentially many coefficients. We later show how the CF representation for analytic functions permits the architecture to be made numan-interpretablee
Output
2
01F +
21
My-
Simply being able to represent a rich class of functions does not imply effective learnability. (After all, unlike Ramanujan, the answer will not simply come to the machine's 's mind.) However, we empirically demonstrate that learning this function class is indeed possible. We propose variants the base architecture catered toward ease of interpretation and efficiency of training, while still minimizing generalization error. We apply CoFrNets to tabular, text, and image data, and show they are either competitive or significantly better than other interpretable models and MLPs. In addition, we are not only able to model synthetic data generated from complicated nonlinear functions accurately, but also obtain feature attributions and recover the functional form reasonably well by leveraging properties of the architecture.
Single ladder (depth d) CoFrNet archi- tecture on the left. On the right the corre- sponding function computed at each stage.
Figure 1:
In summary, the main contribution of this paper new architecture covering an interesting and rich function class. By taking advantage of properties studied from the very beginnings of formal mathematics, we have stumbled upon simple, yet powerful new idea in neural architecture design with much promise for accuracy, interpretabilityy and even efficiency. In this initial paper, we believe we have only scratched the surface of the CoFrNet architecture. Different training strategies and architecture variants as discussed briefly in Sections 4 and 7, among other enhancements, may lead to even better performance in future work.
2 A constant term is assumed to be absorbed in for clearer exposition. 3 See the supplement for the exact quantification of the number of weights.
*Skip connections such as those seen in residual network architectures may end up passing the input to upper layers, but it is unlikely that the input would be consistently passed undisturbed to all upper layers.

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2
Related Work
There are numerous types of explainable machine learning methods. Given our proposal of an interpretable neural architecture, we focus our discussion of related work methods that yield global explanations or interpretable models or neural architectures, even though the latter may be opaque.
Black-Box Neural Architectures. MLP is a standard neural architecture typically composed of fully-connected network [15]. However, MLPs have limitations in their representation and perfor- mance, lcading to many modern architectures. Convolutional neural networks (CNNs) [16] have been very successful in modeling image data. Further improvements were seen with residual network (ResNet) [19] type of architectures which employ ideas such skip connections. This idea is now used in many other architectures such as DenseNet [22] and MobileNetV2 [43]. Transformers [47] have seen immense success for text data and recently were also shown perform well for images [14]. IT-nets [7] were recently shown to perform well for images where they learn a high degree polynomial using tensor decomposition to reduce dimensionality of the search space. Neural networks with activation functions other than the standard ReLU or sigmoid such as those based on Padé approximation [33] have also been suggested. Other architectures such as pi-sigma networks [46] and capsule networks [42] have also found wide appeal.
Globally Interpretable Models. Standard machine learning models such logistic regression, decision trees [5], and rule based models [41, 10] are globally interpretable. Generalized additive models (GAMs) [6] and their more recent variants such as neural additive models (NAMs) [1] and explainable boosted machines (EBMs) [36] also belong to this category. LassoNet [26] is one of the most recently proposed architectures that can be considered interpretable. However, it is restrictive in the sense that if a feature is not selected by itself, it will not appear in any interaction terms. This precludes accurate estimation of functions which have only interaction terms including the very simple bivariate function x T222
Local to Global Post Hoc Methods. There post hoc explanation methods which take local explanations and create global ones. TreeShap [28] creates a global SHAP explanation for tree based models. While, model agnostic multilevel explanation (MAME) [40] can create global LIME explanations. Alternatively, the global Boolean feature learning (GBFL) [37] method leverages local constrastive explanations [13] to create globally interpretable based models.
Self-Explaining Models. Another category of models may not be globally interpretable, but provides local explanations without post hoc mechanisms. [2] is suited for tabular and image data, whereas [53] is suitable for text data. [20] is a framework that provides explanations for new examples explanations are available for training examples. All of these methods, however, do not readily expose the global behavior the model.
3 Preliminaries
We now introduce some notation and discuss equivalent forms for representing continued fractions. We also discuss some of their properties. As mentioned in the introduction, the generalized form for a continued fraction is do
b where a&s and b&s can be complex numbers. If none of a2+
the ag or by are zero Vk E N, then using equivalence transformations [23], one can create simpler the ag = 1 vk E N, with ag 0 in the latter form. A
equivalent forms where cither the by more concise way to write these two forms as follows: i) ag +
= dg + a1 1_ and ii) b b2 Form is known as canonical form. We will interchangeably use the different 1+ 22 ++
forms in the paper based on convenience. One of the nice properties of continued fractions is that in representing any real number with natural number parameters ak, by E N, the rational approximations formed by any of its finite truncations (termed convergents) are closer to the true value than any other rational number with the same or smaller denominator. A continued fraction is therefore the "best" possible rational approximation in this precise sense [23, 31].
In the case where dg and/or by are linear functions of variable E RP, these can be written as functions expressible by power series expansions around æ = O, where there a one-to-one correspondence between the coefficients of both [23, 31]. Hence, given parameter vectors wa E RP,
3

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Figure 2: Three variants of the CoFrNet architecture. In all three variants, the output (top triangle) linear combination of the ladders below a) CoFrNet-F is the full-fledged variant where each ladder receives the whole input x at every stage. b) CoFrNet-D is diagonalized variant where each ladder only receives one of the input dimensions 2j and hence is additive model. c) CoFrNet-DL is a combination of the diagonalized variant and the full variant. The full ladders are of increasing depth and can be understood capture the respective order of interactions.
we can write the following equality as functions of x::
2 +
(1)
W1 W2
for tuples of powers & ..... hp and complex numbers Ci......g We will leverage this relationship in Section as one the strategies to interpret CoFrNets, since it allows us to express a CF as a power series and hence derive feature attributions for individual features well their interactions.
4 CoFrNet Architecture
In this section, we present the proposed architecture based on CFs with three variants. We then discuss aspects of effectively training such architectures and how to obtain feature attributions for interpretation.
Proposed Architecture. We focus continued fractions in canonical form, with unit numerators by = 1. As in (1), we let the denominators dş w x be linear functions of the input æ. Then with fractional function that work with is
denoting the depth, the basic
1 Od
f(x; w)) = 4g +
Ok = w x.
(2)
0d
Each such function corresponds to the diagram Figure 1. We refer to such function as "ladder" due to this pictorial representation with rail and rungs that carry the input to each node.
We propose three variants of the architecture, shown in Figure where each variant is linear combination of functions f in (2), i.e., a combination of ladders. We propose CoFrNet-F as full- fledged variant which all ladders receive the full input at each layer. We propose diagonalized variant CoFrNet-D, in which each ladder operates
single input dimension æjj, i.e., f ())) The linear combination of these ladders is therefore an additive model and directly interpretable [18]. Finally, we propose CoFrNet-DL, which contains both single-feature ladders and full ladders of increasing depth starting at depth two and hence capturing interactions to that order. This variant combines the benefits of the other two architecture variants.
Training Strategies. As discussed in the introduction, we regard the proposed architectures as neural architectures in which the input is passed to all layers, linear functions with weights wy are computed,

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and the nonlinear activation function is the reciprocal 2 -+ 1/2. All variants are differentiable and can thus be trained using standard techniques such as ADMM and backpropagation and popular frameworks such as TensorFlow PyTorch. Other commonly used ideas such dropout [44] could also be leveraged for better generalization.
The most natural choice is to jointly train all the ladders; however, one could envision other training strategies. For example, one could use boosting where one or a group of ladders are first (jointly) trained and where we successively train new ladders on the residuals with appropriate examplc weighting. One could also incrementally fit each layer within ladder. Another strategy might be to collapse the linear combination of ladders into single rational function. However, it may be a challenge to tie the coefficients of this rational function to the weights wig; not constraining the rational function coefficients in this way would result in an exponential number of coefficients to estimate (exponential in the depth of the ladders).
Handling Poles. A key issue that arises from the _ reciprocal nonlinearity is that the denominator may go to zero during training, leading to the function being undefined at that point (a pole in the context of rational functions). To tackle this issuc, we slightly alter the activation function to sgn(z) max([z].e] for some (small) > O, where |- denotes absolute value. The can be fixed to a small positive value or tuned during training. Other solutions to this problem involve either restricting each of the denominators to be positive [33] or the final denominator of the rational function to be positive [3]. Both of these constraints can be restrictive as well as computationally challenging.
Interpretability. We now discuss two strategies to interpret the full-fledged version of our architec- ture CoFrNet-F. Both strategies exploit its functional form. As mentioned, CoFrNet-D an additive model and can be interpreted by visualizing the univariate functions fy (zy; w()) that compose it.
i) Interpretation using Continuants (IC): It is well-known from the theory of continued fractions [23] that (a;; w) in (2) can be expressed as the following ratio of polynomials,
1 f(a; w) = au + a1
1
1 (ao, ..0.,....,.... ad)
(3)
0d-1+ Od Ka(ai...., ,d))
where the polynomials KR, known as continuants, satisfy the recursion
Ko == 1, K₁ (as) Od, K2
(4)
= ad 04-x41Ki-1004--220
ad) +
a.))
(5)
The following result (proven in the supplement) provides compact expression for the gradient of (a; us) with respect to the inputs J. = ,P.
Proposition 1. The partial derivative of f(x; w) with respect to uz is given by
af(x; dz;
(-1)* ( Ka(ai,...,00. 02)
+...,, Wijk-
k=
Proposition 1 provides computationally efficient means to compute the gradient of a ladder with respect to its inputs, which is useful for multiple feature- methods of interpretation [34]. Given an input x, and assuming that the linear functions a* w x: have already been computed in evaluating fæ; w.. the continuants K₂ can be computed using (5) in (d) operations. Then for each input zi.. the sum in Proposition also requires (d) operations, for a total of O(dp) operations for all æy. Proposition udditiinally suggests an interpretation of the coefficients Wikk as contributions to the partial derivative for æy, weighted by ratios of continuants and with alternating signs.
For linear combinations of ladders f as in Figure 2, the above result extends straightforwardly since differentiation is linear operation. This yields feature attributions in (Ldp) time for L ladders.
ii) Interpretation using Power Series (IPS): The above method using continuants gives first-order attributions at a per example level. To obtain higher-order as well as first-order global attributions, we turn to the representation of a ladder in (1) as a multivariate power series, where as mentioned before there is a one-to-one mapping between the coefficients of the two forms. A linear combination of ladders, which our architecture entails, can also be represented by multivariate power series by summing the coefficients Ci.....?, for each monomial term. These coefficient sumss thus provide attributions for individual features T, as well as higher-order interactions, up to the depth of the ladders.
5

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For low-depth ladders, it is possible to manually equate and find the appropriate coefficients based on a linear recurrence relation [39]. However in general, manual computation can be too laborious. For such cases recommend using symbolic manipulation tools such as Mathematica [51]. For a function that is a linear combination of ladders ff of depth d, one can obtain the power series expansion up order dp by applying the following set of Mathematica operations:
N Normal [Series [g. f O, d. ... {Tp, 0, d)l]]
(6)
where "Series" produces a Taylor series expansion, "Normal" implies normalized form, and "N" represents fractional coefficients decimals. The appropriate coefficients can then be picked off to determine feature attributions or attributions for interactions.
5 Universal Approximation
We now prove (Theorem 2) that a linear combination of continued fractions has the property of universal approximation. More preciscly, we show this for the family of functions that are lincar combinations of a finite number of continued fractions, each with finite depth and linear layers. Our strategy essentially comprises three steps: i) showing polynomials of linear functionals (PL) are unital subalgebra [4] and separating on the domain, ii) applying the Stone Weieeseeissss theorem [45] to show that they are thus dense in the space of bounded continuous functions, and iii) showing that PL are subset of the aforementioned class of functions, i.e. finite number of finite-depth continued fractions where each layer is linear function of the input. This implies the latter class is also dense in the space of bounded continuous functions, a.k.a. universal approximators.
Without loss of generality we consider the domain d(x, v) = ||l->>>ll 2112, 2, y É X. Since X bounded and closed, it is compact metric space. The space of bounded continuous functions C( RR) = {f + R f is continuous, EM s.t. [f() S M Vr x) Let (=))) max (z)]. Also for the proof we explicitly mention the constant term for each linear function which was subsumed in in previous sections.
[0, 1, along with the usual Euclidean metric
Definition 1. (Identity Function) Define Id(x) to be the identity function where Id(x) = 1 Va E
Definition 2. P C C(x, R) is unital subalgebra if a) Id(z) P. b) Vf.g E P.f P. c) Vf,geT P and a, 3 R, of + 8g- E P..
Definition 3. P C C(2, R) is separating on the domain X if y E I + v, 3f EP: f(z) + (v).
Theorem 1. (Stone-Wer Weieestraass Theorem [45]] x is compact metric space, and if C C(x, R) is a unital subalgebra and separating on the domain X, then P is dense in C(x, R) with respect to the Loo metric.
Definition 4. (Polynomials of Linear Functions) Define
PC = { + Σ cs
co. cs E IR us E RP Vk E [m], Sc 21m), m E EN; (TT
(7)
SES
KES
to be the set of polynomials on linear functions 3i of
In the above definition, note that we may have it - uy for I + k to obtain higher powers of uf x. Lemma 1. PL a unital subalgebra and is separating on x. PC is dense in C(x.R). R).
Proof. Let f f(x), g(x) E PL. It is easy to see that fx) * g(x) E PL and of + 39(x) E PL for all 8 E R. Further, setting c 1 and cs 0 in the definition of PL yields the identity function. Therefore, PL is a unital subalgebra.
For any two x: + y, V, E X. let u E R2, * 0 be such that it does not belong to the null space of x - y. Such u can always be found by the rank-nullity theorem applied to the subspace
span(x y) in the vector space R., Now, consider f() + u² 8 É PL. Then, f(x) * f()
since u (x y) * 0. This shows that PL is separating the domain X. Hence, by Theorem P.
is dense in C C(x, R).
0

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Definition 5. (Continued Fractions with Linear Functions) Let
0
0g
of 0n 0/2 x: + 02
1/ x. x odd
CFC
1+
+ w/ B1+ 1 w x 32t
1 1+ w / + Dd
Vo, Vk, wk E R², 0n. ag Bk E IR, 15 k d, d d NE
(8)
be the set of finite-depth continued fractions with affine functions u x + or and w x + Bk as numerators and denominators.
In the above definition, we are explicitly writing the constant terms ark, Bg for clarity.
Given a set A of functions on let A + A {aa + Bb a, b E a, E RJ be the set of linear combinations of two functions from A, and + A the set of linear combinations of L functions from A.
Theorem 2. (Representation Theorem) PC C C(x,R).
D' CFC. Also,
Ü L=1
+' CFL #'CFL
is dense in
L=1
Proof.
By Euler's s formula for continued fractions we have:
4n
01
02
Od
ag + a0a1 + ap41
(9)
1+1+ar+
a2
1 ad
By applying the above formula twice to two nested sums, we have:
ao
-01
-02
ad
a0
01
02
Odd-
a041 ... .. = 1+ 1 01 1 a2t
+ Oh 1+ 01 1 a2
1 0d
(10)
Now represent a monomial Lee (4f x). c R. we observe that we need set dg = c, aj (u[a) vk E {l....,d d.. in (10). This in turn can be realized as a member of CFC by setting Up
og c, Wk -tk uk. and Okk Ak 0 for k 1,...,d. d. Hence we have Kekee E CCCLCCC CFL.
Now consider f(x) = cu + Eseses
I1
[ 2) E PC. Then we have f E + 2|
CFC by
RES
doing term by term expansion in terms of CFL CFL. This implies that PL C U D
CFL.
L=1
Combining with Lemma implies that L=1
U + CFCFLis dense in C(x,
Remark. Note that Ü C'CFC does contain functions with singularities in IR. This implies
L=1
that contains functions that are not in C(x,R) But nevertheless, dense C(x, R). The presence of singularities is the reason why is not possible directly apply proof techniques using the Hahn-Ban Bahn-BBnacc theorem [9], which are used for proving representation theorems for two-layer neural networks. It is noteworthy that one only needs linear functions in every layer of continued fraction and linear combinations of these to represent any bounded function on compact set.
Compactness of Representation for Learning Sparse Polynomials. If one wants to learn a sparse polynomial in the variables 4; where the number non-zero monomials |S] is a constant and degree bounded, i.e. |SS| < d, VS E S, Lasso-based techniques would require a representation which pd in size (although sample complexity may be polynomial in d log p) [35]. However,
representation would require only 2|S] + parameterized ladders each of depth at most d. Ellicient learning of sparse polynomials using such compact representations is an interesting direction for future work.
6 Experiments
We now conduct synthetic and real data experiments. The goal of the synthetic experiments two fold: i) to show that can accurately model well known non-linear functions [50] (viz. Matyas

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function, Rosenbrock function, etc.), but more importantly ii) to show that our architecture lends itself to global interpretation using the two strategies described in Section 4. Performance comparisons in terms of mean absolute percentage enror (MAPE) with MLPs of the same depth and with similar number of parameters are in the supplement, as our main intent here is to showcase interpretability.
We then experiment on seven real public datasets covering tabular, text and image data. For six of them in addition MLP we compare against four well known interpretable models namely, GAMs (github. wtt honp///wwa/pyGGM) NAMs (github. com/ni wttpsiittoosssououooooeeeee aaciurallddddtataaa additive seddiiiiemmmesss) _models). EBMs (github. com comm//nttrrprrrcallnntttrrrrrrt and CART decision trees as well as the recent LassoNet (github. com
with the main intent being to compare test accura- cies. For the text data we used Glove embeddings [38]. We report feature attributions for some of the datasets in the main paper.
We average results over five random train/validationntess splits (65%/5%/3309) for all datasets except those that come with their own pre-specified test set. A two P100 GPU system with 120 GB RAM was used to run the experiments. In the activation function, e was set to 0.1 to mitigate poles.
6.1 Synthetic Experiments
In these experiments we use the CoFrNet-F variant to model different synthetic functions, based on a sample of 300 points for each function. We compute feature attributions using the con- tinuants strategy, and the entire function using the power series strategy mentioned in Section 4. A single full ladder is used in each case and the depth is equal to the degree of the func- tion we are approximating. For non-polynomial functions we set the depth to be six. We choose CoFrNet-F since it is the hardest variant to interpret; we show that it can be interpreted.
Figure shows two well-known non- lincar functions: the Matyas func-
tion and the Rosenbrock function.
CoFrNet-F is able to accurately ap- proximate both (7.31% % and 13.08% MAPE respectively). Importantly, the IPS interpretation leveraging the one- to-one correspondence to power series is able to replicate the functions quite closely. The constructed interpreta- tion is not only close in prediction, but also in (univariate) feature attribution. For Matyas, we even recover higher order coefficients accurately (possibly due to a better fit). The linear and constant terms have very small coeffi- cients in the Matyas function approx- imation, making the IPS and original function very similar. Moreover, the feature attributions for the linear terms also recovered by IC, the closed- form formula involving continuants.
100
4x10
50
10
y
-100
-5
100
X
x
100
10
1x
10
100
50
-100
y
y
-5
-100
-5
X
100
X
5
(b)
OF: 0.005 0.005-0.01z
+
(a) OF: 0.54x 0.54y
my.
0.005z
0.5y² 2²y 0.5z
IPS: 0.56I + 0.44y
2y +
IPS: 0.002 0.01x 1 0.008
0.06z 0.07y 0.01.
0.01x"y x 0.09
8).
IC: 0.06 + 2, -0007 + y.
IC: 10.00.01 + 2. 0 +
Figure 3: Original function (OF; in yellow) and the corre- sponding IPS approximation (in blue) for the Matyas func- tion (left) and Rosenbrock function (right). The subfigure caption lists the OF, IPS, and feature attributions from IC. The equations for OF and IPS are normalized by maximum coefficient. Coefficients of the same order terms (that are close) are color-coded coded for ease of comparison. CoFrNet-F is able to approximate the shape of the functions well. The (univariate) feature attributions for IPS and IC are consistent.
6.2 Real Data Experiments
We evaluate our approach relative to other approaches on Credit Card, Magic and Waveform tabular datasets [12], the sentiment analysis [29] and
Quora Insincere Questions [24] text
datasets, and the CIFARI0 [25] image dataset. We also experimented with our approach on the
ImageNet dataset [11]. The dataset characteristics are in the supplement. We report performance of the CoFrNet-DL DL architecture, which was the best performing among the three variants and also
8

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Table Test accuracies different methods on six real datasets. For datasets without pre-specified test sets, a paired t-test was conducted to determine statistical significance. DNC denotes 'did not converge. Best results (within statistical error and excluding SOTA) are in bold.
Methods
Interpretable
Waveform
Magic
Credit Card
CIFAR10
Sentiment
Quora
CoFrNet-DL.
Yes
0.81
0.84
0.69
0.87
0.84
0.88
CoFiNet-D
Yes
0.69
0.76
0.66
0.38
0.80
0.75
GAM
Yes
0.85
0.85
0.72
DNC
0.51
DNC
NAM
Yes
0.86
0.81
0.69
0.38
0,50
0.49
EBM
Yes
0.85
0.85
0.72
0.40
0.59
0.49
CART
Yes
0.75
0.79
0.69
0.29
0.52
0.73
LassoNet
Yes
0.84
0.76
0.67
0.28
0,50
0,53
MLP
No
0.34
0.65
0,50
0.35
0.83
0.85
SOTA
0.86
0.86
0.75
0.99
0.96
0.94
CoFrNet-D to show how well our simplest form performs. For CoFrNet-D, the depth of the univariate ladders varied up 250. For CoFrNct-DL, besides the p univariate ladders, we consider up to 50 full ladders with increasing depth (maximum depth of 50) as per the architecture in Figure 2. We used early stopping. dropout, batching and Adam optimizer with weight decay. For CIFARIO we also did data augmentation (i.e. random cropping and flipping). The best performance for most datasets using CoFrNet-DI. was obtained using ladders of depth 12 or less.
We observe in Table 1, that the CoFrNet-DL model is competitive with other interpretable models on the tabular datasets (i.e. within few percent) and within 6% of the state-of-the-art (SOTA) black-box models for these datasets (gradient boosted trees) [37]. On images and text, although we are farther off from SOTA black-box models ([17] for CIFARIO and [52] for text), we are significantly better than other interpretable models, and similar or better than (uninicrprctable) MLP. We believe this because CoFrNets can compactly represent rich class of functions in nigh-dimensional space where properties of CFs such as fast convergence are likely witnessed. We additionally also trained the CoFrNet-DL model on ImageNet and obtained an accuracy of 0.69, which comparable to ResNet- 18 type architectures.
We showcase the interpretability of our CoFrNet-DL model in Figure 4. Figures 4a, 4b and 4c depict the (functional) behavior of the most important diagonalized ladders in our CoFrNet-DL models trained on the Waveform, Magic and Credit Card datasets respectively. Similar plots for the top three features in cach dataset are provided in the supplement. The important features also seem to make semantic sense given the respective tasks. For example, the Credit Card dataset "Bill statement in April, 2005" should have an impact on predicting payment defaults, since someone not having repaid their previous balance would presumably have higher likelihood of defaulting, Figure 4d, shows the feature attributions for individual images using the IC strategy. In this case global attributions make little sense to report, which is why we report local explanations. Our explanations seem focus on critical aspects such as wings of the plane, the body and face of the horse, and the face and ears of the dog. More such explanations are again provided in the supplement. Figures and 4f depict the most important words, filtering out articles, prepositions and auxiliary verbs, that are highlighted by our Co models on the Sentiment and Quora datasets. It makes sense that words such as "good", "bad" and "like" should play an important role in gauging sentiment, while words such as "some", "people" especially taken together, could indicate sarcasm/insincerity More discussion of these results is provided in the supplement.
Discussion
In this paper, we have proposed CoFrNets, new neural architecture inspired by continued fractions. We have theoretically shown its universal approximation ability, empirically shown its competence on real-world datasets where we are either competitive or much better than other interpretable models as well as MLPs, and analytically shown how to tease interpretability out from it. The training optimization relies on specific properties of CFs: while CFs are rational functions, optimizing rational functions directly leads lo either exponemtially-mmny coeflicients constraints that are difficult to enforce, but these pitfalls do not happen when the CF representation is used. Similarly, interpretation is obtained efficiently and naturally by leveraging the theory of CFs.

---

Plot of Ladder
Plot of fLength Ladder
Bill Statement in April '05 _adder
4.8
Ladder Outour
Ladder Output
Ladder Output
4.6
4.4
0.4
0'5
0.05
0.10
0.15
0,050
0.075 0.100 0.125
X7 Variable Value
1Length Vaniale value
Bill Statement '05 Var Value
(a) Waveform
(b) Magic
(c) Credit Card
airplane
think
worstgreat good can
can that
bes does people
first nollike bad
are
dog
some
you
(d) CIFAR10
(c) Sentiment
(n) Quora
Figure 4: In the first row for the three tabular datasets (a-c), we see the behavior of the most important features indicated by our CoFrNet-DL model. In the second row, first (d) local explanations for CIFAR10 10 using the IC strategy. Shades closer to blue indicate high importance, while those closer red imply low importance (colormap in the supplement). The two figures (e, word clouds highlighting the words our CoFrNet-DL model considers as most important. More example explanations provided in the supplement.
We hypothesize that CoFrNets have favorable adversarial robustness properties due to their functional simplicity and we intend to test this in future work. The hypothesis arises from the following argument. Ladders of small depth, which we find to have good accuracy, yield smooth low-degree rational functions that make it difficult for small input perturbations to produce large changes in output. We may even be able analytically compute adversarial robustness metrics akin to the CLEVER score for CoFiNets [48].
Beyond this initial proposal of empirical accuracies that match or outdo state-of-the-arr black-box models across modalities. Since CoFrNet can be viewed as a neural architecture, we have been able exploit the well-developed tools available to train neural architectures. It is possible however that different training strategies such as those mentioned in Section 4 could be advantageous: each ladder could be built rung by rung. or linear combination of ladders could be built incrementally. Similarly, while the modification of the 1/2 function a practical solution avoid singularities, it is possible that also limits the expressiveness of the function, and more advanced ways to defeat poles [3] could be less restrictive. In addition, known tactics from other neural architectures (convolutional blocks, pooling, maxout) or even something new may be warranted. It is possible that convolutional blocks may have natural implementation based on older filter design literature signal processing that builds upon CFs [32].
architecture, we admit there room for improvement to achieve
Funding and Conflicts of Interest
All authors were employed by IBM Corporation when this work was conducted. There were no other sources of funding.
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References
[1] R. Agarwal, N. Frosst, X. Zhang, R. Caruana, and G. E. Hinton. Neural additive models: Interpretable machine learning with neural nets. In arXiv:2004. 13912, 2020.
[2] D. Alvarez-Melis and T. Jaakkola. Towards robust interpretability with self-explaining neural networks. Advances in Neural Information Processing Systems, pages 7775-7784. 2018.
[3] A. Austin, M. Krishnamoorthyy S. Leyffer, S. Mrenna, Müller, and Schulz. Practical algorithms for multivariate rational approximation. Computer Physics Communicutions, 261, 2021.
[4] N. Bourbaki. Elements of mathematics. Ppringer-Verlag. 1989.
[5] L. Breiman, Friedman, C. J. Stone, and R. Olshen. Classification and Regression Trees. Chapman and Hall, 1984.
[6] R. Caruana, Y. Lou, J. Gehrke, P. Koch, M. Sturm, and N. Elhadad. Intelligible models for healthcare: Predicting pneumonia risk and hospital 30-day readmission. In Proceedings the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD '15, pages 1721-1730, New York, NY, USA, 2015. ACM.
[7] G. Chrysos, S. Moschoglou, G. Bouritsas, Deng, Y. Panagakis, and S. Zafeiriou. Deep polynomial neural networks. In Intl. Conference on Computer Vision and Pattern Recognition (CVPR), 2020.
[8] Churchill, Brown, and Verhey. Complex Variables and Applications. Springer-Verlage 1989.
[9] G. Cybenko. Approximation by superpositions of sigmoidal function. Mathematics of control, signals and systems, 2(4):303-314, 1989.
[10] S. Dash, O. Günlük, and D. Wei. Boolean decision rules via column generation. In Advances in Neural Information Processing Systems, pages 4655-4665 2018.
[11] J. Deng, W. Dong, R. Socher, L.J. Li, K. Li, and L. Fei-Fei. Imagenet: A large-scale hierarchical image database, In 2009 IEEE conference on computer vision and pattern recognition, pages 248-255, Icee, 2009.
[12] D. Dheeru and E. Karra Taniskidou. UCI machine learning repository, 2017.
[13] A. Dhurandhar, P.-Y. Chen, R. Luss, C.-C. Tu, P. Ting, K. Shanmugam, and P. Das. Explanations based on the missing: Towards contrastive explanations with pertinent negatives. In Advances in Neural Information Processing Systems, pages 592-603, 2018.
[14] A. Dosovitskiy, L. Beyer, A. Kolesnikov, D. Weissenborn, X. Zhai, T. Unterthiner, M. De- hghani, M. Minderer, G. Heigold, Gelly, Uszkoreit, and N. Houlsby. An image is worth 16x16 words: Transformers for image recognition at scale. In Ind. Conference on Learning Representations (ICLR), 2021.
[15] Fehollet, Matsuyamax, and Kemaswill Keras MNIST MLP implementation. In
2017.
[16] Fehollet, Matsuyamax, Smerity, and Kemaswill. Keras MNIST CNN implementations. In https://github..e https://gihaab.co/fcchlllete www.///nnillleeee/////eeeeeeeaee pom. 2017.
[17] Foret, A. Kleiner, H. Mobahi, and B. Neyshabur, Sharpness-aware minimization for efficiently improving generalization. In Inil Conference on Learning Representations (ICLR), 2021.
[18] J. Hastie and R. J. Tibshirani. Generalized Additive Models. Chapman and Hal//CRC, 1990.
[19] K. Hc, X. Zhang, S. Ren, and Sun. Deep residual learning for image recognition. In Intl. Conference Computer Vision and Pattern Recognition (CVPR), 2015.
[20] M. Hind, D. Wei, M. Campbell, N. C. F. Codella, A. Dhurandhar, A. Mojsilovic, K. N. Rama- murthy, and K. R. Varshney. TED; Teaching AI to explain decisions. In Proceedings of the AAAI/ACM Conference on AI, Ethics, and Society, pages 123-129, 2019.
11

---

[21] K. Hornik. Some new results on neural network approximation. Neural networks, 608):1069- 1072, 1993.
[22] G. Huang, Z. Liu, L. Maaten, and K. Q. Weinberger. Densely connected convolutional networks. In 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 2261-2269, 2017.
[23] W. B. Jones and W. Thron. Continued fractions. fractinn, Analytic theory and applications. Encyclopedia of Mathematics and its Applications. Addison
Addicoo Addison-Wesley. 1980.
[24] Kaggle. Quora insincere question dataset, 2019.
[25] A. Krizhevsky. Learning multiple layers of features from tiny images, 2009.
[26] I. Lemhadri, L. A. Feng Ruan, and R. Tibshirani. LassoNet: A neural network with feature sparsity. In AISTATS, 2021.
[27] Z. Lu, H. Pu, F. Wang, Z. Hu, and L. Wang. The expressive power of neural networks: A view from the width. In Advances in Neural Information Processing Systems, 2017.
[28] S. M. Lundberg, G. Erion, H. Chen, A. DeGrave, J. M. Prutkin, B. Nair, R. Katz, J. Himmelfarb, N. Bansal, and S.-I. Lee. From local explanations to global understanding with explainable ai for trees. Nature Mach. Intl., 2020.
[29] A. L. Maas, R. E. Daly, P. T. Pham, D. Huang, A. Y. Ng, and C. Potts. Learning word vectors for sentiment analysis. In The 49th Annual Meeting of the Association for Computational Linguistics (ACL), 2020.
[30] Mathologer.
How Ramanujan solved the Strand 2020.
puzzle.
[31] K. Milton. Summation techniques, Padé approximants, and continued fractions. 2011. http: //www. wwn..nnnnnoo.. ou. www..nh..ou...uuuuiiiooe wat.edditttt///t511..... thap8.pdd.
[32] S. Mitra and R. Sherwood. Canonic realizations of digital filters using the continued fraction expansion. IEEE Transactions Audio and Electroacoustics, 20(3):185-199) 1972.
[33] A. Molina, P. Schramowski, and K. Kersting. Padé activation units: End-to-end learning of flexible activation functions in deep networks. In Intl. Conference on Learning Representations (ICLR), 2020.
[34] C. Molnar. Interpretable Machine Learning. L https///chiiishhhngghhhhiiihihrr 2019.
[35] S. Negahban and D. Shah. Learning sparse Boolean polynomials. In 2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pages 2032-2036, IEEE, 2012.
[36] H. Nori, S. Jenkins, P. Koch, and R. Caruana. InterpretML: A unified framework for machine learning interpretability. In arXiv:1909.009
2019.
[37] T. Pedapati, A. Balakrishnan, K. Shanmugam, and A. Dhurandhar. Learning global transparent models consistent with local contrastive explanations. In Advances in Neural Information Processing Systems, 2020.
[38] J. Pennington, R. Socher, and C. D. Manning. Glove:
Global vectors for word representation. In Empirical Methods in Natural Language Processing (EMNLP), pages 1532-1543, 2014.
[39] M. W. Pownall. Functions and Graphs: Calculus Preparatory Mathematics. Prentice-Hall -Hal, 1983.
[40] K. N. Ramamurthy, B. Vinzamuri, Y. Zhang, and A. Dhurandhar. Model agnostic multilevel explanations. In Advances in Neural Information Processing Systems, 2020.
12

---

[41] C. Rudin. Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead. Nature Mach. Intell., 163):206-215, May 2019.
[42] S. Sabour, N. Frosst, and G. E. Hinton. Dynamic routing between capsules. In Advances in Neural Information Processing Systems, 2017.
[43] M. Sandler, A. Howard, M. Zhu, A. Zhmoginov, and L.-C.. Chen. MobileNetV2: Inverted residuals and linear bottlenecks. In 2018 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 4510-4520, 2018.
[44] N. Srivastava, G. Hinton, A. Krizhevsky, Sutskever, and Salakhutdinov. Dropout: A simple way to prevent neural networks from overfitting. Journal of Machine Learning Research, 15(56):1929 1958, 2014.
[45] M. Stone. Applications of the theory of Boolean rings to general topology. Transactions of the American Mathematical Society, 41(3):375-455 -8,,1, 1937.
[46] H. Swapna Rekha, J. Nayak, and H. S. Behera. Pi-sigma neural network: Survey of decade progress. In A. K. Das, J. Nayak, B. Naik, S. Dutta, and D. Pelusi, editors, Computational Intelligence in Pattern Recognition, pages 429 441, Singapore, 2020. Springer Singapore.
[47] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, u. Kaiser, and I. Polosukhin. Attention is all you need. In I. Guyon, U. V. Luxburg. S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, Advances Neural Information Processing Systems, volume 30. Curran Associates, Inc., 2017.
[48] T. Weng, H. Zhang, P.-Y. Chen, J. Yi, D. Su, Y. Gao, C.-J. Hsieh, and L. Daniel, Evaluating the robustness of neural networks: An extreme value theory approach. In Proceedings of the International Conference on Learning Representutions, 2018.
[49] Wikipedia. Coffer, 2021.
[50] Wikipedia. Test functions for optimization, 2021.
[51] Wolfram Research. Wolfram language function, 2020.
[52] Z. Yang, Z. Dai, Y. Yang, Carbonell, R. Salakhutdinov, and Q V. Le. XLNet: Generalized autoregressive pretraining for language understanding, In Advances in Neural Inf. Processing Systems, 2019.
[53] M. Yu, S. Chang, Y. Zhang, and T. S. Jaakkola. Rethinking cooperative rationalization: Intro- spective extraction and complement control. In EMNLP, 2019.
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Reviews:
[{"review_id": "TaphVUrraPF", "reviewer": "Reviewer_bQJD", "summary": "The authors propose three variants of a novel neural architecture based on continued fractions, where the coefficients are learned linear combinations of features.\nThe authors claim global intepretability of this architecture through the additive nature of one variant and introduce two approaches exploiting the architecture's analytical form to compute first and higher order feature attributions.\nThey provide an original theoretical proof that a linear combination of continued fractions, i.e. their architecture, is a universal approximator.\nFinally, the authors demonstrate the performance and interpretability of the architecture on a 2-dimensional toy experiment, as well as on multiple real-world data sets compared to other models.", "questions": "", "limitations": "", "rating": 8, "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": "### Originality:\nContinued Fractions are well-known, and to the best of my knowledge, this is the first time they were suggested for designing a neural architecture.\nAs far as I can tell, the interpretability the authors provide by its functional form has not been done like this before.\nWhile I must admit that I had difficulties following the proof for universal approximation, it is original as far as I can tell.\n\n### Quality:\nThe quality of this paper is quite high, and the work feels well polished.\nThis work motivates the introduced neural architecture well, and provides not only a wide range of experiments to justify its existence, but also proves its property as a universal approximator.\nThe authors discuss weaknesses of their architecture.\n\nSince the authors put a strong emphasis on the interpretability aspect of their architecture, I find the demonstration thereof somewhat short-coming in the manuscript.\nGiven the strong emphasis, I would have liked to see a more quantitative analysis of the interpretability aspect, especially compared to the other globally interpretable models.\n\n\n### Clarity:\nI find the manuscript is well written and well organized.\nBeyond technicality, the manuscript is relatively easy to follow.\nMethods are introduced adequately.\n\n### Significance:\nThe architecture is a new approach for globally interpretable models, based on the well-known continued fractions.\nAlthough the manuscript is limited in its practical demonstration of the architecture's iterpretability,\nI believe it provides a foundation for equally interesting research and demonstrates a new instance to prove universal approximation.\n\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Some limitations are discussed with respect to with the architecture's practical applicability.\nSocietal impact is left unaddressed in the paper.\nA very short remark in Section 7 to its (and other globally interpretable neural network architectures) impact may be beneficial.\n", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "5"}, "extra_text": {}}, {"review_id": "SDL1yg4XEB8", "reviewer": "Reviewer_hnfL", "summary": "The paper proposes CoFrNet, a neural architecture that stacks neural layers in a way inspired by Continuous Fractions. The paper shows some interpretability advantage of the proposed model.", "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": "Pros:\n1. There is some novelty as this is (to my best knowledge) the first work that combines Continuous Fractions with Neural Networks.\n2. Theoretical development on the interpretability via Power Series and universal approximation seems sound.\n3. The paper is in general clearly written.\n\nCons:\n1. While the paper does show the interpretability advantages on synthetic functions and some of the datasets tested, discussion on other datasets ( such as Cifar-10) is missing. \n2. I also feel that the interpretability discussion on non-synthetic datasets is very limited. The authors only listed a few top features. I think a more thorough analysis, possibly including plots illustrating the feature attributions, is needed.\n3. The performance results (in term of accuracy) are poor compared to SOTA models. While the proposed CoFRNet does have interpretability advantages, the poor performance will not allow it to be used in any real-world scenarios. I also doubt if CoFRNet can be applied to more difficult datasets (such as ImageNet) as the inverse activation will have serious gradient variance issues when you stack many more layers to build a deeper model.\n\n\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Limitations are discussed adequately. I don't find any discussion on societal impact but I guess there is no need.\n\n", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "3"}, "extra_text": {}}, {"review_id": "9_dHY4CTSVh", "reviewer": "Reviewer_ByEC", "summary": "The paper introduces CoFrNet, a new neural net architecture inspired by continued fractions. Several different training strategies are suggested, such as joint training, incrementally fit each layer, etc. Synthetic data and real data experiments are provided, and accuracy comparisons compared to several interpretable generalized additive models, trees, neural nets, etc. are provided", "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": "The ideas presented in this paper are interesting. I would like to clarify some questions about the evaluation: \n-\tAre the numbers in Table 1 accuracies or AUCs? The caption says accuracies but just wanted to confirm. If some of these datasets are imbalanced, accuracies would not be a good measure. \n-\tWhy does MLP have such a low accuracy on Waveform (0.34), a lot lower than the other methods where the lowest accuracy was 0.69?\n-\tThat the GAM and EBM results are exactly the same for the 3 tabular datasets (Waveform, Magic, Credit Card) seems a little unusual to me because the pygam package uses different types of spline base learners, if I’m not mistaken, while EBM uses a tree base learner, and previous work in the literature (e.g. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.433.8241&rep=rep1&type=pdf which first introduced EBMs) showed that the choice of base learner can matter a lot. \n-\tPlease provide standard deviations for Table 1 so that the reader can tell if the results are statistically significant.\n\nSince interpretability is one of the goals here, it would be nice to provide some visualizations of the output that humans can interpret. It appears that CoFrNet-D can be visualized as an additive model (one feature at a time), while CoFrNet-F and CoFrNet-DL cannot. However CoFrNet-D seems to have worse performance than other additive models like EBM, NAM, etc.\n\nWriting: \n- “Reference [2] is suited for tabular and image data, 112 whereas [52] is suitable for text data. Reference [19] is a …”  seems a little non-standard to say Reference [x], maybe use author names, e.g. name et al. …..?", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "I don't have any concerns here.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "1"}, "extra_text": {}}, {"review_id": "6nEMxn5h3MW", "reviewer": "Reviewer_Sp1r", "summary": "The authors propose a new neural architecture inspired by continued fractions, which has some great theoretical properties in terms of function approximation. As a neural architecture, CoFrNets can learn function with much less number of parameters compared to fully connected neural networks. The authors also claims that CoFrNets is interpretable since it can represented as a polynomial.", "questions": "", "limitations": "", "rating": 8, "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": "The idea of considering continued fractions for neural networks is novel, however, the claim that the model is interpretable is not supported well. In (1), the authors discussed first-order attributions and high-order attributions. Arguably, if one wants to get a sparse linear/ polynomial function, learning linear models on polynomial expansions and kernel methods would be what comes to mind. I would argue that these two have at least the same explainability to CoFrNets, but they are not compared to.\n\nSome theoretical discussion may also be added compared to polynomial functions. It is mentioned that the rational approximations formed by any of its finite truncations (termed convergents) are closer to the true value than any other rational number with the same or smaller denominator. However, this is not useful as a function approximator. If the Ground Truth is a high order polynomial of order k with d terms, a polynomial expansion would need to represent the function with (# features)^k input features with d non-zero term. Thus, the number of parameter here is  (# features)^k, and the actual parameter needed is d. In this case, can you prove the number of parameters needed to accurately estimate the polynomial function? It seems like you definitely need less than (# features)^k parameters (although I don't know if polynomials are a subset of continuous fractions). Some analysis will also be helpful to readers unfamiliar with continued fractions.\n\nTo raise the quality of the work, I would consider 1. convolution CoFrNets, where each ladder only receives the input dimensions that are structurally similar. 2. using CoFrNets in existing NN layers, so that it is comparable with SOTA models. If CoFrNets is only replacing the most memory-heavy component of some existing model, it might make the model more lightweight.\n\nThe model architecture kind of reminds me of transformer models, and by the self-attention mechanism, it would be much easier to construct polynomial functions for transformers. However, I have doubts that whether continuous fractions is better at learning functions that are not polynomial (see sin cos example in appendix). I feel like the authors did not do enough empirical studies to understand why CoFrNets is better or worse compared to MLP. Universal Approximation already exists for MLPs, and NNs are known to be notoriously over-parameterized (which is one of the reasons that it succeeds), and thus outperforming NNs with the same number of parameters is not really impressive. I would like to understand what is the function space where CoFrNets performs better, and what is the function space where CoFrNets performs better. For example, in image classification CNN will perform better than CoFrNet even if CoFrNet has the same number of parameters compared to CNN (at least for now). If for any function CoFrNets outperforms MLP with \"any'' number of parameters (not just the parameters cherry picked for CoFrNets), I will be very surprised. I would like to see the performance of CoFrNet and MLP both varying the number of parameters (instead of only one set of parameters which is probably cherry picked for CoFrNets). Also, I would like to see more function classes being compared to. For example, you can set the ground truth to be some random forest regression output, and compare how well MLP and CoFrNet compare since most function spaces (estimated in practice) are not polynomials (at least not with sparse parameters). Discussing and understanding when CoFrNet performs worse than MLP (and also when it is better) is going to make the paper more valuable .\n\n--------------------------------------------------------------------------------\n\nAfter the authors showed results for imagenet I have faith for this method to become very impactful in the future. This is a very clear accept in my opinion. The interpretability claim is not needed to make this a strong work as the potential of the new model architecture is very high. More theories and applications can be developed along this type of research.\n\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "yes, but not enough, see main review", "review_text": "", "extra_scores": {"ethical_concerns": "no", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "5"}, "extra_text": {}}]

Author rebuttal:

Meta-review: The reviewers have agreed on many positive aspects of the paper and the authors have done a great job in addressing the concerns. The paper is on a highly important topic and contains novel ideas. 

There were many important points in the reviewer responses that should be integrated into the final version of the paper. 
Decision: Accept (Poster)