Prototype Based Classification from Hierarchy to Fairness
Authors: Mycal Tucker
Julie Shah
Venue: ICLR 2022 Submitted (2022)
Keywords: prototypes
fairness
hierarchy
neural network
encoding

Abstract: Artificial neural nets can represent and classify many types of high-dimensional data but are often tailored to particular applications -- e.g., for ``fair'' or ``hierarchical'' classification. Once an architecture has been selected, it is often difficult for humans to adjust models for a new task; for example, a hierarchical classifier cannot be easily transformed into a fair classifier that shields a protected field. Our contribution in this work is a new neural network architecture, the concept subspace network (CSN), which generalizes existing specialized classifiers to produce a unified model capable of learning a spectrum of multi-concept relationships. We demonstrate that CSNs reproduce state-of-the-art results in fair classification when enforcing concept independence, may be transformed into hierarchical classifiers, or may even reconcile fairness and hierarchy within a single classifier. The CSN is inspired by and matches the performance of existing prototype-based classifiers that promote interpretability.

Under review as a conference paper at ICLR 2022
PROTOTYPE BASED CLASSIFICATION FROM HIERAR- CHY TO FAIRNESS
Anonymous authors Paper under double-blind review
ABSTRACT
Artificial neural nets can represent and classify many types of high-dimensional data but are often tailored to particular applications c.g., for "fair" or "hierar- chical" classification. Once an architecture has been selected, it is often difficult for humans to adjust models for a new task; for example, hierarchical classifier cannot be easily transformed into fair classifier that shields protected field. Our contribution in this work is new neural network architecture, the concept sub- space network (CSN), which generalizes existing specialized classifiers to produce a unified model capable of learning spectrum of multi-concept relationships. We demonstrate that CSNs reproduce state-of-the-arr results fair classification when enforcing concept independence, may be transformed into hierarchical classifiers, or may even reconcile fairness and hierarchy within a single classifier. The CSN is inspired by and matches the performance of existing prototype based classifiers that promote interpretability.
INTRODUCTION
Neural networks are able to learn rich representations of data that support highly accurate classi- fication; however, understanding or controlling what neural nets learn remains challenging. Some techniques offer insight into pre-trained models by uncovering directions within latent spaces that correspond to particular concepts, image manipulations, or more (Goetschalckx et al.) 2019 Kim al. 2018), while approaches focused on interpretability provide techniques that are more com- prehensible to humans (Li et al. 2018 Chen et al., 2019). While these methods provide insight, they fail to offer control: humans observe learned patterns but are unable to guide models such that learned relationships are useful for particular setting or task.
Another line of work has advanced the design of models for particular types of classification tasks (such as fair or hierarchical classification) but these techniques are often developed with only one problem in mind (Zemel et al., 2016, Xie et al. 2017, Hase et al.) 2019). For example, models built for fair classification (predicting an outcome regardless of information about protected field) are only used to enforce independence of concepts rather than hierarchy. Thus, humans may exert control over learned representations by selecting appropriate technique rather than tuning training parameters within the same technique.
We have designed new neural network architecture, the concept subspace network (CSN), which generalizes existing specialized classifiers to produce a unified model capable of learning a spectrum of multi-concept relationships. CSNs use prototype-base
representations, technique employed in
interpretable neural networks in prior art (Li et al. 2018 Chen et al., 2019) Garnot & Landrieu order to simultaneously learn multiple concepts;
2020). A single CSN uses sets of prototypes classification within single concept (e.g., "type of animal") is performed by projecting encod- ings into concept subspace defined by the prototypes for that concept (e.g.. "bird," "dog." etc.). Lastly, CSNs use measure of concept subspace alignment to guide concept relationships such as independence or hicrarchy.
In experiments, CSNs performed comparably state-of-the art in fair classification, despite prior methods only being designed for this type of problem. In applying CSNs to hierarchical clas- sification tasks, networks automatically deduced interpretable representations of the hierarchical problem structure, allowing them to outperform of-te-of-the-rn for a given neural network back- bone, in terms of both accuracy and average cost of emrrs the CIFAR 100 dataset. Lastly, in

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numan-motion prediction task, we demonstrated how single CSN could enforce both fairness (to preserve participant privacy) and hierarchy (to exploit a known taxonomy of tasks). Our find- ings suggest that CSNs may be applicd to wide range of problems that had previously only been addressed individually, or not at all.
2 RELATED WORK
2.1 INTERPRETABILITY AND PROTOTYPE NETWORKS
Numerous post-hoc explanation techniques fit models to pre-trained neural nets; if humans under- stand these auxiliary models, they can hypothesize about how the neural nets behave (Ribeiro et al.) 2016 Lundberg & Lee, 2017). However, techniques in which explanations are decoupled from un- derlying logic may be susceptible to adversarial attacks or produce misleading explanations (Heo) ct al. 2019 Slack al. 2020).
Unlike such decoupled explanations, interpretability research seeks to expose a model's reasoning. In this work we focus on prototype-based latent representations in neural nets. There is long history of learning discrete representations in continuous spaces, originating under "vector quantization" lit- erature (Kohonen 1990 Schneider et al. 2009). More recently, the prototype case network (PCN) comprised an autoencoder model that clustered encodings around understandable, trainable proto- types, with classifications made via a linear weighting of the distances from encodings to prototypes (Li et al. 2018). Further research in image classification extended PCNs to use convolutional filters as prototypes and for hierarchical classification in the hierarchical prototype network (HPN) (Chen et al. 2019: Hase et al., 2019). Lastly, Garnnt & Landrieu (2020) use prototypes in Metric-Guider Prototype Learning (MGP) conjunction with loss function to cluster prototypes to minimize user-defined costs.
Our model similarly uses trainable prototypes for classification, but differs from prior art in two re- spects. First, modify the standard PCN architecture to support other changes, without degrading classification performance. Second, like HPNs (but not PCNs or MGP), CSNs leverage multiple sets of prototypes to enable hierarchical classification but also allow for non-hierarchical concept relationships.
2.2 FAIR AND HIERARCHICAL CLASSIHICATION
AI fairness research considers how to mitigate undesirable patterns or biases in machine learning models. Consider the problem of predicting a person's 's credit risk: non-causal correlations between age and risk may lead AI models to inappropriately penalize people according to their age (Zemel jet al., 2016). The problem of fair classification is often framed as follows: given inputs, æ, which informative of protected field, s, and outcome, y, predict y from without being influenced by s (Zemel et al.] 2013). Merely removing s from x (c.g., not including age as an input to a credit predictor) rarely removes all information about 8, so researchers have developed variety of techniques to create representations that "purge" information about s (Zemel et al.) 2016)] Xie et al.,
2017 Jiang al. 2020).
Hierarchical classification solves a different problem: given a hierarchical taxonomy of classes (e.g.. birds vs. dogs high level and sparrows vs. toucans low level), output the correct label each classification level. Neural nets using convolution and recurrent layers in specialized designs have achieved remarkable success in hierarchical image classification (Zhu & Bain 20017 Guo al., 2018). The hierarchical prototype network (HPN) uses prototypes and a training routine based upon conditional subsets of training data to create dierarchically-organie
prototypes (Hase al. 2019)
Gamot & Landrieu (2020) also use prototypes for hierarchical classification in Metric-Guiided Pro- totype Learning (MGP) by adjusting the training loss to guide prototype arrangement. Neither HPN nor MGP explicitly models relationships between multiple subsets of prototypes. Lastly, recent works propose hyperbolic latent spaces as a natural way to model hierarchical data (Dai et al. 2021 Mathieu et al. 2019; Nickel & Kielal 2017| Liu et al. 2020). Our method, conversely, relies upon concepts from Euclidean geometry. Extending the principle of subspace alignment that we develop to Con-Fuclidean geometric spaces is promising direction but beyond the scope of this work.
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3 TECHNICAL APPROACH
In this section, we outlined the design of the CSN, which was inspired by desires for both inter- pretable representations and explicit concept relationships. First, we wished for interpretable rep- resentations, so we built upon the PCN design, with modifications. Second, explicitly encoded relationships between concepts by introducing multiple sets of prototypes, instead of just one in PCNS. Third, enabled guidance of the concept relationships by modifying the CSN training loss. Together, these changes supported not only interpretable classification, but also provided a flexible framework for single model architecture learn different concept relationships.
3.1 CONCEPT SUBSPACE CLASSIFICATION
A CSN performing a single classification task (e.g., identifying a digit in an image) is defined by three sets of trainable weights. First, an encoder parametrized by weights 8, cg. maps from inputs of dimension X to encodings of dimension Z: R + R Second, a decoder parametrized by weights c, dæ. performs the decoding function of mapping from encodings to reconstructed inputs: de R² R* Third, there exist set of & trainable prototype weights, p, that are each Z- dimensional vectors: P1, P1,P2, Pk E R This architecture resembles that of the PCN, but without the additional linear classification layer (Li et al. 2018).
Here, we focus briefly on the set of prototypes, p. Given set of l prototypes in R we define "concept subspace," C as follows:
U = P1
Vi [2,
(1)
C fx|x E R where
P1 +
Azufoor X E R Vi}
(2)
FE|2,
C' is the linear subspace in R² defined by starting at the first prototype and adding linear scalings of vector differences all other prototypes. We call this subspace a concept subspace because represents a space of encodings between prototypes defining single concept (e.g., prototypes for digits 0, 1, 2, etc. define a concept subspace for digit classification).
A CSN's architecture consisting of an encoder, a decoder, and a set of prototypes and the as- sociated concept subspace - enables two types of functionality: the encoder and decoder may be composed to reconstruct inputs via their latent representations, and CSNs may perform classification tasks by mapping an input, x, to one of Y discrete categories. Classification is performed by first encoding input into a latent representation, then calculated, yielding & distance values: dz(2, p) = ||z psl 2 E [1, k:.. These distances mapped probability distribution, Px (i) [1, k). by taking the softmax of their negatives. Lastly, if there are more prototypes than classes, (e.g., two prototypes for dogs, two for cats, etc.) the distribution over & is converted to distribution over Y categories summing the probabilities for prototypes belonging to the same class.
eo (x).. The 12 distance from to each prototype
For single-concept classification, CSNs differ from PCNs primarily by removing the linear layer that PCNs used to transform distances to prototypes into classifications. We found this unnecessary for high classification accuracy (Appendix A) and instead directly used negative distances. Without the linear layer, CSN classification is equivalent to projecting encodings, onto concept subspace before calculating distances. The distances between projected encoding, dubbed žproý: and proto- types will induce the same softmax distribution as when the orthogonal component remains. Indeed, we find projection more intuitive only the component of that corresponds the subspace used for classification and list projection example projecting an encoding and calculating distances to prototypes is shown in Figure I a.
standard step in the remainder of this paper. A simple
For some tasks, we used an encoder design from iitatiionllautoencoode (VAEs) in order to reg- ularize the distribution of encodings to conform to unit Gaussians (Kingma & Welling 2014). By default, this regularization loss was set to O, but it sometimes proved useful in some domains to prevent overfitting (as detailed in experiments later). We emphasize that CSNs are discriminative, rather than generative, models, did not seek to learn latent space from which to sample.
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2
Z
Pio
P1
P:h
Serth
Ptt
proj
P2
P3
P21
(d)
Figure I: CSNs support single (a) or multiple (b-d) classification by projecting encodings into con- cept subspaces. Subspaces may exhibit range of relationships from parallel (b) orthogonal (d).
3.2 MULTI-CONCEPT LEARNING
We defined the CSN architecture for single classification tasks in the previous section; here, plain how a CSN may be used for multiple classification tasks. (For example, consider a scenario involving classifying both what type of bird photo depicts and whether the photo was taken out- doors or indoors.) Extending CSNs to support multiple classifications requires the addition of new sets of prototypes. This the primary contribution of our work.
Multiple classification tasks are performed by defining a set consisting of sets of prototypes: prototypes for each of classification tasks. classification task is
P = (P1 ...,...,,, with performed by using the CSN's encoder to generate an encoding, and projecting into the concept subspace defined by the set of prototypes particular to the given task. Figure [ (b-d) depicts sim- plified examples of two concept subspaces. In each example, each concept uses three prototypes, yielding two planar concept spaces (one of which corresponds to the x - y plane for illustrative purposes); may be projected into either plane depending upon the classification task at hand.
While the prototypes in different sets are separate from each other, correlations present in training data may lead to range of relationships among prototypes. Returning to the previous example scenario, prototypes of birds may represent canaries and toucans, while prototypes of indoor and outdoor scenes may represent living rooms and jungles; each set of prototypes is independent in principle, but in reality, prototypes may represent canaries in living rooms and toucans in jungles. In fact, two sets of prototypes can exhibit range of relationships from highly correlated to fully independent, as shown in Figure
We defined metric, concept subspace alignment, to reflect this range of relationships. Mathemati- cally, the alignment two subspaces the mean of the cosine squared of the angle between all pairs of vectors drawn from the basis of each subspace. Given orthonormal bases, efficiently computed via R factorization, Q1 and Q2, of ranks m and n, we define alignment as follows:
a(Q1, Q2)
mn Σ
1
m n
Σ (QT Q2(1,)1)²
Given the range of values for the cosine squared function, alignment values range from 0 to for orthogonal and parallel subspaces, respectively. Intuitively, orthogonality lends itself well to independent concepts and therefore supports fair classification, whereas parallel subspaces naturally correspond to hierarchical classification. We elaborated on this intuition in Section 3.4
3.3 TRAINING PROCEDURE
When training CSN, assume access to a set of training data, (X, for Y (Y,, Yc). For each entry in the dataset, there is an input x and a label yr, for each of e classification tasks.
We trained CSNs in an end-to-end manner to minimize single loss function, defined in Equation The four terms in the loss function were as follows: 1) reconstruction error; 2) the loss introduced for the PCN, encouraging classification accuracy and the clustering of encodings around prototypes (applied within each concept subspace); 3) a KL divergence regularization term; and 4) term penalizing alignment between concept subspaces. Each term weighted by choice of real-
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valued Às We emphasize that the PCN loss clustering and classification accuracy, defined in Equation 7 of Li et al. (2018) - is calculated within each concept subspace using the projections encouraged cluster around prototypes only along dimensions
of encodings; thus, encodings within the subspace. The encoder, decoder, and prototype weights were trained simultaneously.
Ao
1(,, 1X,Y,0, ,,0,,,
Σ (dg(ea(r)) - x)² FEX
+
Σ
Xr PCNPrppprseeo
xE[1.C
+
Σ XRI,KL(X,Pi)
(4)
<E[[c]
Ai JEp
Σ
E
The KL regularization term mimics training losses often used in VAEs that penalize the divergence between the distribution of encodings and a zero-mean unit Gaussian (Kingma & Welling 2014). In our case, we wished to induce a similar distribution of encodings, but centered around prototypes rather than the origin. Furthermore, rather than induce Gaussian distribution within concept subspace (which would dictate classification probabilities and therefore potentially worsen classifi- cation accuracy), we wished to regularize the out-of-subspace components of encodings.
Concretely, we implemented this regularization loss in three steps. First, we computed the orthogo- nal component of an encoding 2orth the distribution of 2orth and unit Gaussians centered at each prototype in each subspace. Finally, we distances between encodings and prototypes in order only select the closest
Zproj- We then computed the KL divergence between
took the softmax prototype to the encoding; we then multiplied the softmax by the divergences to enforce that encod- ings were distributed as unit Gaussians around the nearest prototype in each subspace. Together, these operations led the distributions out of each subspace to conform to unit Gaussians around each prototype. confirmed in later experiments, this component was crucial in training fair classifiers.
3.4 HIERARCHICAL AND FAIR CLASSIFICTION
We conclude this section by demonstrating how CSNs may support hierarchical or fair classifica- tions. Hierarchical and fair classification may be thought of as extremes along spectrum of concept alignment. In hierarchical classification, concepts are highly aligned and therefore parallel: the dif- ference between toucan and Dalmatian is similar the difference between generic bird and dog, and so the vector differences between prototypes associated with different classes should also be parallel (e.g., "bird" "dog" "toucan" "Dalmatian.") In fair classification, concepts aligned: switching belief about someone's 's sex should not alter predictions about their income. Thus, based on the classification task, moving an encoding relative to one subspace should either affect (for hierarchical) or not affect (for fair) that encoding's projection onto the other subspace. We provide a geometric interpretation of these two tasks Figure b and d.
CSNs can be trained to adopt either form of concept relationship by penalizing or encouraging concept subspace alignment (already present as a(Q:, Q,) in the training loss). Our single model reconciles these two types of problems by viewing them as opposite extremes along spectrum of concept relationships that our technique able to learn; this is the main contribution of our work.
4 RESULTS
Our experiments were divided in four parts. First, we demonstrated how CSNs matched standard performance on single classification tasks: in other words, that using a CSN did not degrade perfor- mance. We omit these unsurprising results from the paper; full details included in Appendix A Second, we showed that CSNs matched state-of-the-an performance in two fair classification tasks. Third, used CSNs for hierarchical classification tasks, exceeding performance demonstrated by
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Table I: Mean Adult dataset fairness results.
Table Mean German dataset fairesss results.
Model
y Acc.
Acc.
D.I.
DD-0.5
Model
Acc.
Acc.
D.I.
DD-0.5
CSN
0.85
0.67
0.83
0.16
CSN
0.73
0.81
0.70
0.10
Adv.
0.85
0.67
0.87
0.16
Adv.
0.73
0.81
0.63
0.10
VFAE
0.85
0.70
0.82
0.17
VFAE
0.72
0.81
0.47
0.23
FR Train
0.85
0.67
0.83
0.16
FR Train
0.72
0.80
0.55
0.16
Wass. DB
0.81
0.67
0.92
0.08
Wass DB
0.72
0.81
0.33
0.02
Random
0.76
0.67
Random
0.70
0.81
Acult Encodings
Table 3: Fairness ablation results on the German dataset. Alignment enforced causal independence, and additionally including the KL loss enforced distribu- tional independence. All std.
0.02.
-02
Une trscess
Loss
y Acc.
D.I.
DD-0.5
KL. + Align
0.73
0.70
0.10
0.00
Align
0.73
0.69
0.11
0.00
Figure 2: 2D latent space for fair classifi- cation. The prototypes for high and low in- come (X) are perpendicular the prototypes for applicant sex (circles).
KL
0.72
0.67
0.12
0.20
Neither
0.75
0.64
0.13
0.13
prior art along several metrics. Fourth, we showed how CSNs enabled both fair and hierarchical classification in dataset describing human motion in an assembly task that exploited hierarchical knowledge while preserving participant anonymity. Implementation details of CSNs in all experi- ments are included in Appendix
4.1 FAIR CLASSIFICATION
We evaluated CSN's performance in fair classification tasks in the Adult and German datasets. These datasets are commonly used in fairness literature and contain data that can be used to predict people's income credit risks (Dua & Graff 2017). compared CSN performance to our implementations of an advesarial purging technique (Adv.), the variational fair autoencoder (VFAE), Wasserstein fairness approach (FR Train) (Xie eet al. 2017 Zemel et al. 2016 Jiang et al., 2020, Roh et al., 2020)). Implementation details of fair classification baselines and full results including standard deviations are included in Appendix G
Fair Classification (Wass. DB), and mutual-in iantaaaiiiioooaton-naaaaae
For the Adult dataset, the protected attribute was sex, and for the German dataset, the protected attribute
binary variable indicating whether the person evaluation, we measured Acc., the accuracy of predicting income or credit, s Acc., the accuracy of linear classifier trained predict the protected field from the latent space, disparate impact (DI), as defined in Roh et al. (2020). and demographic disparity (DD-0.5), defined by Tiang et al. (2220).
older than 25 years of age. In
Mean results over 20 trials for both datasets were included in Tables 1 and In both datasets, we observed that CSNs matched srate-of-the-att performance. CSNs produced high y Acc., indicating high task performance for predicting income or credit. Furthermore, fairness measures demonstrate that CSNs purged protected information successfully (low Acc.) and achieved high D.I. and low DD-0.5, as desired. A visualization of the latent space of a fair classifier, trained on the German dataset, is shown in Figure and confirmed that CSNs learned orthogonal concept subspaces.
In addition to reproducing the state of the art, we conducted an ablation study to demonstrate the importance of terms in training loss: the alignment and KL losses. Using the German dataset, we trained 20 CSNs, setting the KL, alignment, or both loss weights to 0. The mean results of these trials reported in Table
Table 3 demonstrates the necessity of both KL and alignment losses to train fair predictors (with higher disparate impact and lower demographic disparity values). Including both loss terms resulted

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in the fairest predictors; removing those losses could enable better classification accuracy, but at the expense of fairness. This confirms geometric intuition: the alignment loss created orthogonal sub- spaces and the KL regularization created distributional equivalence based on the subspaces. Jointly, these losses therefore produced statistical independence.
Table also includes causal analysis of trained CSNs via the P metric. Intuitively, this metric reflected the learned correlation between s and y: it was calculated by updating embeddings in the CSN latent space along the gradient of and recording the change in prediction over y. We reported the ratio of these changes as p; as expected, enforcing orthogonality via alignment loss led to values of 0. This technique is inspired by work in causally probing language models (e.g.. Tucker iet al. (2021)); full details for calculating p are included in Appendix D
4.2 HIERARCHICAL CLASSIFICATION
We compared CSNs to our implementation of HPNs and results for Metric-Guided Prototype Learn- ing (MGP), reported by Garnot & Landrieu 2020), for hierarchical classification tasks. Our HPN baseline used the same architecture as CSN (same encoder, decoder, and number of prototypes). It differed from CSNs by setting alignment losses to and by adopting the conditional probability training loss introduced by Hase et al. (2019). We further included results of randomly-initalizzed CSN under "Init." in tables. In these experiments, we sought to test the hypothesis that CSNs with highly aligned subspaces would support hierarchical classification, just as orthogonal subspaces en- abled fair classification.
In addition to standard accuracy metrics, we measured two aspects of CSNs trained on hierarchical tasks. First, we recorded the "average cost" (AC) of errors. AC is defined as the mean distance be- tween the predicted and true label in graph of the hierarchical taxonomy (e.g., if true and predicted label shared a common parent, the cost was 2; if the common ancestor was two levels up, the cost was 4, etc.) (Garnot & Landrieu 2020). Second, we measured the quality of trees derived from the learned prototypes. After a CSN was trained, we defined a fully-connected graph G = (V, E) with vertices V = IP U{0} (the set of all prototypes and a point at the origin) and undirected edges between each node with lengths equal to the 12 distance between nodes in the latent space. We re- covered the minimum spanning tree, T, from G, (which is unique given distinct edge lengths, which we observed in all experiments), and converted all edges to directed edges through a global ordering of nodes. Lastly, we calculated the graph edit distance (ED) between isomorphisms of the recovered tree and the ground-truth hierarchical tree (with edges that obeyed the same ordering constraints) (Abu-Aisheh et al. 2015). Intuitively, this corresponded to counting how many edges had to be deleted or added to the minimum spanning tree to match the taxonomy tree, ignoring edge lengths, with a minimum value of for perfect matches.
As basic test of CSNs in hierarchical classification tasks, we created simple hierarchies from the MNIST Digit and Fashion datasets. The Digit dataset used the standard low-level labels of digit, supplemented with high-level labels of parity (two classes); the Fashion dataset used the standard low-level labels for item of clothing, with a ternary label for a high-level classification of "tops" (t- shirts, pullovers, coats, and shirts). "shoes" (sandals, sneakers, and ankle boots), or "other" (trousers, dresses, and bags).
Mean results from 10 trials for both MNIST datasets were included in Tables 4 and The HPN baselines were implemented using the same number of prototypes as the CSNs being compared against. Both tables show that CSNs exhibit comparable or better accuracy than HPNs for both the low-level (Y)) and high-level (Y1) classification tasks. In addition, the average cost (A.C.) and edit distance (E.D.) values show that CSNs recovered minimum spanning trees that nearly perfectly matched the ground truth tree, and that when CSNs did make errors, they were less "costly" than errors made by HPNs (although admittedly, a dominant force in A.C. classification accuracy alone). A 2D visualization of the latent space of a CSN trained on the Digit task is shown in Figure 3: encodings for particular digits clustered around prototypes for those digits (X), while prototypes for even and odd digits (circles) separated the digit clusters into the left and right halves of the latent space. Visualizations of latent spaces for more fair and hierarchical classification tasks are included in Appendix C they confirmed the theoretical derivations of orthogonal and parallel subspaces.
Lastly, we trained 10 CSNs and HPNs on the substantially more challenging CIFAR100 dataset. The dataset is inherently hierarchical: the 100 low-level classes are grouped into 20 higher-level
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Table 4: MNIST digit hierarchy mean (stdev)
Table 5: MNIST fashion hierarchy mean (stdev)
over 10 trials. First two columns x 100.
over 10 trials. First two columns x 100.
Yo% %
Y1 %
A.C.
E.D.
%0%
Y, %
A.C.
E.D.
CSN
98 (0)
99 (0)
0.06 (0.00)
3,4 (4.3)
CSN
88 (0)
98 (0)
0.28 (0.01)
0.0 (0.0)
HPN
96 (0)
98 (0)
0.12 (0.01)
23 (1.0)
HPN
88 (0)
98 (0)
0.28 (0.01)
24 (0.9)
Init.
10 (0)
50 (1)
2.80 (0.01)
20 (0.0)
Init.
10 (0)
33 33)
3.13 (0.02)
30 (0.8)
Digit Encodings
Table 6: CIFARI00 hierarchy results using the standard two-level hierarchy (top half) MGP's 5-level hierarchy (bottom half).
X
1,%
Y1
A.C.
E.D.
CSN
0.76 (0.0)
0.85 (0.0)
0.76 (0.02)
11.2 (7)
Figure 3:
2D latent space for hi-
HPN
0.71 (0.0)
0.80 (0.0)
0.97 (0.04)
165.0 (3)
erarchical digit classification creates
Init.
0.01
0.05
3.88
200
clusters around even and odd proto- types (circles on the right and left, re- spectively) and digit prototypes (X).
CSN
(0.0)
0.88 (0.0)
0.91 (0.0)
(8.2)
MGP
0.76
1.05
Init.
0.01
0.05
7.33
258
classes, each of size 5. Using a resnet18 encoder, pre-trained on ImageNet, in conjunction with 100 prototypes for low-level classification and 20 for high-level, we trained CSNs and HPNs. CSNs additionally used an alignment loss weight of -10 to encourage parallelism between the two concept subspaces. The mean results over 10 trials are shown in the top half of Table
We also compared CSNs to MGP and other hierarchical classifiers using the CIFARIOP 100 dataset and deeper hierarchy, consisting of levels of sizes 100, 20, 8, 4, and as done by Garnot & Landrieul 2020). The additional information provided by this deeper hierarchy resulted in improved classi- ficatiin performance. Median results (as done by Garnot & Landrieu (2020)) for 10 CSNs using this dataset are shown in the bottom half of Table 6 Changing the hierarchy changed how average cost was calculated, so values from the top and bottom halves of the table should not be compared. Within the bottom half, we note that CSNs outperformed MGP on both A.C. and classification accu- racy. Furthermore, according to values generated the extensive experiments conducted by Garnot & &Landrieu (2020). CSNs outperformed numerous other baselines, including HXE and soft-labels [Bertinetto et al. (2020), D. YOLO (Redmon et al. (2016)), and a hyperspherical prototype network Mettes al. (219), all of which were built upon a resnet18 18 pretrained on ImageNet. In fact, our CSNs achieve SOTA classification accuracy for any classifier built upon resnet18 backbone, with- out data augmentation. Furthermore, the decrease in A.C. is especially surprising given that other techniques explicitly optimized for average cost reductions, while CSNs merely trained on classifi- cation at each level. Notably, the decrease in A.C. is not fully explained by the increase in accuracy, indicating that CSN not only exhibited higher accuracy but also, when it did make mistakes, those mistakes were less severe.
Lastly, we note that CSNs support range of learned relationships other than fair or hierarchical. The varying values of p in inble(inniccute that CSNs may learn different relationships when align- ment loss is set to O. However, in general, one could train models to learn desired relationships by penalizing or rewarding alignment relative to some intercept. We trained and evaluated such models in Appendix [ and found that models indeed learned the desired alignment.
4.3
FAIR AND HIERARCHICAL CLASSIFICATION
Prior experiments demonstrated how CSNs could solve different classification problems separately; in this section, we applied a single CSN to a task that required to use both fair and hicrarchical classification. Intuitively, faimesss was used to protect privacy, while hierarchical structure was used for better performance.
We used dataset describing human motion in a bolt- plat-peac task. The dataset was gathered from a similar setup to (Lasota et al. 2014) motion was recorded at 50 Hz, using the 3D location of each of the 8 volunteer participant's gloved right hands they reached towards one of 8 holes arranged
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Table 7: CSNs for fair and hierarchical classification on bolt task. Mean (std. dev.) over 10 trials.
Fair
Hier
Bolt%
p
A.C.
D.I.
DD-0.5
No
No
0.81 (0.01)
0.36 (0.07)
2.7 (0.05)
0.80 (0.02)
0.13 (0.02)
Yes
No
0.41 (0.07)
10 10 1003)10000
2.8 (0.10)
0.93 (0.03)
0.05 (0.02)
Yes
Yes
0.55 (0.06)
10-3 (< 10 9)
2.4 (0.05)
0.92 (0.02)
0.06 (0.02)
in a line to place a bolt in the hole. The bolt holes may be thought of hierarchically by dividing destinations into left vs. right (LR) groupings, in addition to the label of the specific hole.
Initial exploration of the dataset showed promising results for prototype-based classification: the target locations were identified with 81% accuracy, and further analysis showed that prototypes corresponded to human-like motions (details in Appendix [B). Troublingly, however a trained CSN could identify the participant with over 60% accuracy, which posed privacy concerns. Nevertheless, from a robotic safety perspective, it is important for robots to exploit as much information as possible to avoid collisions with humans.
Ultimately, we wished to predict which hole a participant was reaching towards given the past second of their motion, while preserving privacy and exploiting hierarchical structure. Thus, designed a CSN with three concept subspaces: one for predicting the bolt (8 prototypes), one for predicting the high-level grouping of left or right destination (2 prototypes) and one for predicting the participant id (8 prototypes). We enforced that the bolt and LR subspaces were parallel while the bolt and participant subpsaces were orthogonal.
Results from our experiments are presented in T Means and standard deviations over 10 trials for each row are reported. (A.C. was calculated only for prediction errors, due to the large differences in accuracy rates across models.) When training CSN with no constraints on subspace alignment, we found a highly accurate but unfair predictor (81% accuracy for bolt location, but sub-optimal disparate impact and demographic disparity values). Switching the CSNs to be fair classifiers by only enforcing orthogonality between bolts and participants yielded a fair classifier (illustrated by p. D.I., and DD-0.5). but with much worse bolt prediction accuracy (44%). However, by using hicrarchical subspace for LR groupings, the final CSN both improved classification accuracy and decreased the average cost of errors, while maintaining desired fairness characteristics.
5 CONTRIBUTIONS
The primary contribution of this work is a new type of model, the Concept Subspace Network, that supports inter-concept relationships. CSNs' design, motivated by prior art in interpretable neural network models, use sets of prototypes to define concept subspaces in neural net latent spaces. The relationships between these subspaces may be controlled during training in order to guide desired model characteristics. Critically, archical classification - are located at either end of a spectrum of concept relationships, allowing CSNs to solve each type of problem in manner on par with techniques that had previously been designed to solve only one. Furthermore, a single CSN may exhibit multiple concept relationships, as demonstrated in a privacy-preseriing hierarchical classification task.
note that two popular classification problems - fair and hier-
While we have demonstrated the utility of CSNs within several domains, numerous extensions could improve their design. First, the idea of subspace alignment could be applied non-Euclidean ge- ometries like hyperbolic latent spaces that are sometimes used for hierarchical classification. Sec- ond, CSNs could additionally benefit from relaxation of some simplifying assumptions: notably, allowing for more complex relationships rather than those defined by subspace cosine similarity, or using adversarial approaches for distributional regularization rather than only supporting unit Gaussians.
Lastly, we note that CSNs, while designed with ethical applications such as fair classification in mind, may lead to undesired consequences. For example, malicious actors could enforce unde- sirable concept relationships, or simply observing emergent concept relationships within a CSN could reinforce undesirable correlations. In addition, although prototypes encourage interpretabil- ity, which we posit can be used for good, the reductive nature of prototypes may be problematic when classifying human-related data (e.g., the COMPAS fair classification task we avoided).

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REFERENCES
Zeina Abu-Aisheh, Romain Raveaux, Jean- Yves Ramel, and Patrick Martineau. An Exact Graph Edit Distance Algorithm for Solving Pattern Recognition Problems. In 4th International Con- ference Pattern Recognition Applications and Methods 2015, Lisbon, Portugal, January 2015. doi: 10.52000005299900010073...
URLhttpa a ouvertes. fr/
ha1-01168815
Luca Bertinetto, Romain Mueller, Konstantinos Tertikas, Sina Samangooei, and Nicholas A Lord. Making better mistakes: Leveraging class hierarchies with deep networks. In Proceedings the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 12506-12515 2020.
Chaofan Chen, Oscar Li, Daniel Tao, Alina Barettt, Cynthia Rudin, and Jonathan K Su. This looks like that: deep learning for interpretable image recognition. In Advances in neural information processing systems, pp. 9930-8941, 2019.
Shuyang Dai, Zhe Gan, Yu Cheng. Chenyang Tao, and JJ (Jingjing) Liu. Apo-vae: Text generation in hyperbolic space. In 19th Annual Conference of the North Amer- ican Chapler of the Association for Computational Linguistics (NAACL 2021), March 2021. URL https:: wttp:/wwwww...oooooo wttpc/ooooot.cco/eeeooeo/ote http:///eeeeaaacccbbbbbblee. apo vae text www.rggeeeinnnnn in -egerbotoooooo---poppooooooo---- space/
Dheeru Dua and Casey Graff. UCI machine learning repository, 2017. httpptt:://acchie
ics. wci.ccc.. uci.edu/ml
Vivien Sainte Fare Garoo and Loic Landrieu. Leveraging class hierarchies with metric-guided prototype learning, 2020.
Lore Goetschalckx. Alex Andonian, Aude Oliva, and Phillip Isola. Ganalyze: Toward visual defi- nitions of cognitive image properties. In Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), October 2019.
Yanming Guo, Yu Liu, Erwin M Bakker, Yuanhao Guo, and Michael S Lew. Cun-rnn: A large- scale hierarchical image classification framework. Multimedia Tools and Applications, 77(8): 10251-10271, 2018.
Peter Hase, Chaofan Chen, Oscar Li, and Cynthia Rudin. Interpretable image recognition with hierarchical prototypes. In Proceedings of the AAAI Conference on Human Computation and Crowdsourcing, volume pp. 32 40, 2019.
Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recog- nition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp- 770-778, 2016.
Juyeon Heo, Sunghwan Joo, and Taesup Moon. Fooling neural network interpretations via adversar- ial model manipulation. In Advances in Neural Information Processing Systems, pp. 2925-2936, 2019.
Ray Jiang, Aldo Pacchiano, Tom Stepleton, Heinrich Jiang, and Silvia Chiappa. Wasserstein fair classification. In Uncertainty in Artificial Intelligence, pp. 862-872. PMLR, 2020.
Been Kim, Martin Wattenberg, Justin Gilmer, Carrie Cai, James Wexler, Fernanda Viegas, et al. Interpretability beyond feature attribution: Quantitative testing with concept activation vectors (tcav). In International conference machine learning. pp.
PMLR, 2018.
Diederik P. Kingma and Max Welling. Auto-encoding variational bayes. In Yoshua Bengio and Yann LeCun (eds.), 2nd International Conference Learning Representations, ICLR 2014, Banff. AB, Canada, April 14-16, 2014, Conference Track Proceedings, 2014. aollattt://aaae /ttp://xxxx/////////////
abs/ 131 6114
Teuvo Kohonen. Improved versions of learning vector quantization. In 1990 ijenn international joint conference Neural networks, pp. 545-550. IEEE, 1990.
10

---

Under review as a conference paper at ICLR 2022
P. A. Lasota, G. F. Rossano, and J. A. Shah. Toward safe close- proximity human-robe interaction with standard industrial robots. In 2014 IEEE International Conference on Automation Science and Engineering (CASE), pp. 539-344, 2014. doi: 10.109//CAAEE2214..8999999
O. Li, H. Liu, C. Chen, and C. Rudin. Deep learning for case-based reasoning through prototypes: A neural network that explains its predictions. In AAAI, 2018.
Shaoteng Liu, Jingjing Chen, Liangming Pan, Chong-Wan Wah Ngo, Tat-Seng Chua, and Yu-Gang Jiang. Hyperbolic visual embedding learning for zero-shot recognition. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 9273 9281, 2020.
Scott M Lundberg and Su-In Lee. A unified approach to interpreting model predictions. In Advances in neural information processing systems, pp. 4765-4774, 2017.
Emile Mathieu, Charline Le Lan, Chris J Maddison, Ryota Tomioka, and Yee Whye Teh. Contin- uous hierarchical representations with poincaré variational auto-encoders. In Advances in neural information processing systems, pp. 12565-12576, 2019.
Pascal Mettes, Elise van der Pol, and Cees Snoek. Hyperspherical prototype networks, In Advances in Neural Information Processing Systems, pp. 1487-1497, 2019.
Maximillian Nickel and Douwe Kiela. Poincaré embeddings for learning hierarchical representa- 2017.
tions. In Advances in neural information processing systems, pp.
Joseph Redmon, Santosh Divvala, Ross Girshick, and Ali Farhadi. You only look once: Unified, real-time object detection. In Proceedings of the IEFE conference on computer vision and pattern recognition, pp. 779-788, 2016.
Marco Tulio Ribeiro, Sameer Singh, and Carlos Guestrin. "Why should I trust you?" explaining the predictions of any classifier. In Proceedings of the 22nd ACM SIGKDD international conference on knowledge discovery and data mining, pp. 1135-1144, 2016
Yuji Roh, Kangwook Lee, Steven Whang, and Changho Suh. FR-train: A mutual information- approach to fair and robust training. In Hal Daumé III and Aarti Singh (eds.), Proceedings of the 37th International Conference on Machine Learning, volume 119 of Proceedings of Machine Learning Research, pp. 8147-8157, PMLR, 13-18 Jul 2020. URL http: htttp://proceedingg: mlr. wwwsss/vl9//eteee.e...eee
html
Petra Schneider, Michael Biehl, and Barbara Hammer. Adaptive Relevance Matrices in Learn- ing Vector Quantization. Neural Computation, 21 21(12):3532-3561, 12 2009. ISSN 0899-7667. doi: wttp///ccc.200.11--000 908. URL https: http::::///oo...//////// wrg/10 www2//ecccccoo 200................. 2009.
11 08-908
Dylan Slack, Sophie Hilgard, Emily Jia, Sameer Singh, and Himabindu Lakkaraju. Fooling lime and shap: Adversarial attacks on post hoc explanation methods. In Proceedings of the AAAI/ACM Conference on AI, Ethics, and Society, AIES '20 pp. 180-186, New York, NY, USA, 2020. Association for Computing Machinery, ISBN 9781450377 100. doi: 10. 10.11453375622.3335553 URL https::/doiii mwtoi.orgg/000 1145/ 000555337562..7 3375833
Mycal Tucker, Peng Qian, and Roger Levy. What if this modified that? syntactic interventions with counterfactual embeddings. In Findings of the Association for Computational Linguistics: ACL-IJCNLP 2021, pp. 862-875, Online, August 2021. Association for Computational Linguis- tics. doi: 10.18653/v1/202 1/255////1222.... w.....innnn-aade 76.. URL httph: httt::://aalaatthlloyy org./221... Firdings-acl..7 76
Qizhe Xie, Zihang Dai, Yulun Du, Eduard Hovy, and Graham Neubig. Controllable invariance through adversarial feature learning. In Advances in Neural Information Processing Systems, pp- 585-596, 2017.
Rich Zemel, Yu Wu, Kevin Swersky, Toni Pitassi, and Cynthia Dwork. Learning fair representations. In International Conference on Machine Learning, pp. 325-333, 2013.
Yujia Li Max Welling Richard Zemel, Christos Louizos, and Kevin Swersky. The variational fair autoencoder. In Proceedings the International Conference on Learning Representations, 2016.
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1
2
3
4
5
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7
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Figure 4: Decoded prototypes for digit classification provide visualizations of how images are clas- sified.
Xinqi Zhu and Michael Bain. B-cnn: branch convolutional neural network for hierarchical classifi cation. arXiv preprint arXiv: 709.09899, 2017.
A SINGLE-CONCEL CLASSIFICATION BASELINES
In addition to the specialized fair and hierarchical classification tasks, we tested CSNs on two stan- dard classification tasks: identifying digits in the MNIST Digit dataset, and identifying one of 100 categories in the CIFAR100 dataset. There were no concept relationships present because this was a single classification task; instead, the tests established that using CSN did not degrade classifica- tion performance relative to PCNs or other neural architectures.
On the Digit dataset, trained CSN using the same encoding and decoding layer architectures with 20 prototypes (two for each digit) and applied the same Gaussian distortions to training images as [i et al. (018)). Over five trials, training for 50 epochs with batches of size 250, achieved the same mean classification accuracy as PCN (9.,22%) demonstrating that the use of CSN did not worsen classification accuracy (Li et 2018).
On the CIFARI00 dataset, we extended a resnet18 backbone (pretrained on ImageNet) as our en- coder with 100 prototypes, for each class, and trained 10 models for 60 epochs He et al. (2016). We achieved a mean classification accuracy of 76%, the standard result for networks built upon resnet18 framework (Hase et al.) 2019). Thus, CSNs exhibited high performance in challenging domain, matching performance of normal networks, with the benefit of interpretable prototypes.
B VISUALIZING DECODED PROTOTYPES
Because CSNs are built upon prototype- based classification, they are at least as interpretable as prior art, such as PCNs. In this section, demonstrate how prototypes may be decoded visualize their representations. These figures were generated by decoding prototypes from the models used throughout the paper.
Figure 4 shows the first 10 prototypes from the MNIST digit classifier in Appendix A Unlike PCNs, CSNs benefit from an inductive bias that leads to an equal number of prototypes per class.
Figure depicts the decoded prototypes when training CSN to predict human reaching motions, as described Section 3.4 This model was only trained on motion prediction, without using fair or hierarchical training terms. Interestingly, by weerparametiizing the number of prototypes (there were only 8 possible destinations, but twice as many prototypes), the model Iearned different forms of trajectories that reached towards the same destination: short movements near the targets, and longer loops when reaching from farther away.
C VISUALIZING LEARNED LATENT SPACES
In addition to decoding prototypes, we visualized the latent spaces of trained classifiers. For the purposes of visualization, we trained models from scratch, using only 2D latent spaces. En- codings and prototypes for both fair classification tasks, as well as the digit and fashion hierarchical classification tasks, are shown in Figure
In all diagrams, encodings of test inputs are denoted by small colored dots. All classification tasks used 2 sets of prototypes: depieted one set of prototypes large black dots, and the other as X's. The arrangement of the prototypes in the latent spaces confirms that CSNs have learned the right concept alignment.
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0.8
Label
0.6
0.4
0.2
0.6
0.5
0.0
0.4
Label
0.2
0.4
0.3
X Label
0.6
0.8
0.2
1.0
Figure 5: Decoded prototypes for the bolt classification task, plotted within an axis-normalized 3D space. Using 16 prototypes for the 8-label classification task reveals bolt-specific actions as well as a variety of motions such as screwing in the bolt and moving towards
Specifically, for the fair classification tasks, the X's form line segment that is orthogonal to the line formed by black dots. This orthogonality leads to fairness, as discussed in our paper.
In the hierarchical domains, similarly observed that CSNs had learned the "right" latent structure. In these domains, the black dots denoted prototypes for high-level classification (such as even vs. odd). We observed that the lower-level prototypes (e.g., for digit), denoted by 'X's were clustered around the high level prototypes.
As whole, these visualizations confirm that CSNs learn the desired latent structure, all controlled by changing the alignment loss weight.
D CALCULATING LEARNED CASUAL RELATIONSHIPS
Sections 4.1 and 4.3 we report metric, p. to denote the learned causal relationship between concepts. Here, we explain how we calculate p in greater detail.
Intuitively, p corresponds to the mean change in belief for one classification task divided by changes in belief for another classification task. For example, in fair classification, prediction of a person's 'ss credit risk should not change based on changes in belief over the person's 's age; this notion corre- sponds p O.
We calculate in CSNs using a technique inspired by Tucker et al. (2021). In that work, the authors studied if language model's output changed when the model's 's internal representation changed according to syntactic principles. In our work, we change latent representations, 2, by taking the gradient of with respect the loss of one classification task given true label y". creating new z!
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Adult Encodings
German Encodings
04
High Incame
Red
tew Inname
Goad Credz
-0.4
Digit Encodings
Fashion Encodings
Antle boot
Bag
Sneaker
Sandal
Dress
Trouser
-4
T-shin/tpp
2.0
2.0
Figure 6: 2D learned latent spaces for the Adult, German, Fashion, and Digit datasets, showing prototypes from two concepts (x 's and black dots). Fair tasks used orthogonal concept spaces (top); hierarchical tasks created multi-level clusters (bottom).
taken by moving along that gradient, and then calculating the new classification likelihoods using z'.
For simplicity, we limit our analysis to CSNs with two concept subspaces. We denote the encoding of an imput, x, as z = e (x). Prediction for each of the two tasks may be denoted as prediction functions, pred,, for i E [0, 1] indicating the two tasks; the prediction function corresponds to projecting & into the relevant subspace and calculating distances to prototypes, as discussed earlier. Using this notation, define formally:
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German Encodings
Weather Encodings
2.0
15-
Credi
PARYY
0.0
Good Credz
-15
X
+0.30 -0.5
0:30 0.75
2.
20
Figure 7: Without alignment loss guidance, CSNs may learn undesirable relationships between con- cepts like age (circles denote prototypes) and credit (Xs denote prototypes) in the German dataset (left). In an artificial weather scenario (right), CSNs can be guided to learn the right causal relation- ship between temperature and precipitation.
2 = cσ(x)
(5)
yu pr)d
(6)
91 = p (
(7)
= = + Vloss(yo..6)
(8)
1/ = pred
V = pr |
(10)
P Yo 36
(11)
1/1 vÍ
Thus, P captures how the model's change in belief about one attribute affects its change in belief' over another attribute, in other words the causal learned relationship between prediction tasks.
E CORRELATED-CONCEER CLASSIFICATION BASELINES
A single CSN may be used to perform multiple classification tasks simultancously without explicitly guiding concept relationships. In the Adult and German fair classification domains, CSNs predicted both s, the protected field, and the desired final prediction, while we explicitly guided the learned concept relationships to enforee fairness. In a separate set experiments conducted on the same datasets, we demonstrated how CSNs can learn more complex concept relationships.
In these experiments, we trained CSNs with two subspaces, each with two prototypes, and set both the KI. and alignment losses to zero. The CSNs were trained to predict both and y, using the two subspaces. We recorded prediction accuracy of y and p, the learned causal correlation between and y.
Over 10 trials, for the German and Adult datasets, CSNs achieved mean y classification accuracies of 85% and 74%, par with prior art these datasets when not enforcing fairness (Xie al.) 2017). We also found non-zero p: for the German dataset, we found a value 0,20; for the Adult dataset, mean value of 0.23. An example latent space from CSN trained the German dataset in this manner is shown in Figure / using the same visualization mechanism as introduced in Appendix C In this example, the model learned non-zero correlation between prototypes for credit (Xs) and
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for applicant's is age (cireles). This type of learned correlation is undesirable in fair classification domains but may be useful in other scenarios.
As demonstration of useful learned correlations, we implemented CSN in classification task us- ing synthetic data. Consider a simplified weather prediction task in which, given noisy observations of temperature and precipitation, weather station must classify the day hot or cold and rainy or sunny. In the artificial world, in the last year of weather data, half of the days are rainy and half are sunny, and all rainy days are cold and all sunny days are hot. Cold days have a true temperature uni- formly drawn between 0.0 and 0.2 and warm days have a true temperature uniformly drawn between 0.8 and 1.0. Similarly, sunny days have a precipitation value drawn uniformly between 0.0 and 0.2 and rainy days have precipitation values are drawn uniformly between 0.8 and 1.0. Observations of temperature and precipitation are corrupted by zero-mean gaussian noise with a 0.05. Given noisy observations of temperature and precipitation, model's 's task is to predict binary labels for whether the day is hot or cold and rainy or sunny.
Numerous causal paths could explain observational data recorded from this environment in which hot days are sunny and cold days are rainy: rain could cause cold weather, some latent factor like atmospheric pressure could affect both precipitation and temperature, etc.. Trained simply from observational data, models are unable to learn the right causal relationship between these variables.
Unlike traditional neural networks, however, CSNs allow humans to encode desirable causal rela- tionships. We designed a CSN with two concept subspaces (for temperature and precipitation), each with two prototypes. We then penalized (a(Qrs: Qhe) vp-)² ) that is, we for the alignment loss between the two subspaces for rainy and sunny (rs) and hot and cold (hc). (We used p as the notation for setting desired alignment for reasons that will become apparent in the next paragraph.)
intercept of yp"
In our experiments, we sought to identify if CSNs could learn the desired causal relationship between temperature and precipitation. We did so by setting v0 to some value, training CSNs using standard losses, and then measuring if the p metric we calculated from the trained CSNs matched p².
We trained 10 CSNs with latent dimension 2 with vo = 0.5. This corresponds to a cosine value of 0.7, or about 45 degrees. This is intuitively interpreted as meaning that for every percentage increase in likelihood in the weather being sunny, the likelihood of it being warm should increase by 0.7 percent.
As desired, the trained CSNs had mean P value of 0.72 (standard deviation 0.14). An example latent space from one such CSN shown in Figure the two subspaces are arranged roughly 45 degrees. Moving an embedding of cold and rainy day to increase the likelihood of it begin sunny by 1% increases the predicted likelihood of it being warm by 0.7%. This demonstrates that we were able to train CSNs to learn the causal relationship we wished for. Lastly, we note that we repeated these experiments training loss, CSNs learned arbitrary concept relationships.
e with other vp" and found similar findings, and if we did not include alignment
alignmenn
F CSN IMPLEMENTATIONS
In this section, we include details necessary for replication of experimental results that did not fit in the main paper.
In all experiments, we used random seeds ranging from 0 to the number of trials used for that experiment. Although CSNs support several prototypes per class (e.g.. prototypes for the digit O, 2 prototypes for the digit 1, etc.), unless otherwise noted, we used an equal number of prototypes and classes.
In the German fairness experiments, we trained for 30 epochs, with batch size 32, with classification loss weights of 1, alignment loss of 100 between the two subspaces, and KL lossses of 0.5. In the Adult fairness experiments, we trained for 20 epochs, with batch size 128, with classification loss weights of [1. 0.1] for y and s, respectively. alignment loss of 100 between the two subspaces, and KL losses of 0.1. For both fair classification tasks, the CSN encoders comprised 2 fulle-connected layers with ReLu activations and hidden dimension 128, outputting into a 32-dimensional latent
Anonymized code is available at http:////ammggooo.....oooscccccs
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space, and the decoder comprised 2, 128 188dimmnsional ReLu layers followed by a sigmoid layer. The whole model was trained with an Adam optimizer with default parameters.
For the digit and fashion hicrarchical classification tasks, CSNs were trained for 20 epochs with batch size 128. CSNs for both datastes used identical architectures to the networks created for the fair classification tasks.
For the CIFARI00 hierarchical classification task, we built upon a ResNet18 backbone, as done by Gamno & Landrieu (2020). The encoder consisted of ResNet encoder, pre-trained on imagenet, followed by two fully-connected layers with hidden dimension 4096, feeding into a latent space of dimension 100. The decoder (and decoding training loss) was removed in this domain reduce training time. The network was trained using an SGD optimizer with learning rate 0.001 and mo- mentum 0.9, with batch size 32. Training terminated after 60 epochs or due to early stopping, with patience of 10 epochs. Classification loss weights were set to [1, 5] for classifying the high- and low- level categories, respectively. KL losses were set to 0; alignment loss was set to to encourage parallel subspaces.
For the fair and hierarchical classification task with bolts, CSNs used the same architecture as for the fair classification task. Models were trained for 50 epochs, with batch size 256. All classification loss weights were set to 1; alignment loss between bolts and LR was set to -10 and between bolts and participant was set to 100. KL losses were set to 2.
G FAIR CLASSIFICATION BASELINES
In Section 4.1 compared CSNs to several fair classification baselines the standard German and Adult datasets. Although the datasets are standard in the literature, there are a wide variety of fairness metrics, only subset of which each method has published. Therefore, we implemented each fair classification baseline and recorded all metrics of interest for each method. In this sec- tion, demonstrated the soundness of our implementations by comparing to published metrics. Implementations of our baselines are available here: https://anonymous. wttp:/oowwmoooooooooooooooeeee scinncessacnnn baselines-44A7
Tables g and report the recorded metries for each fairness technique. Values reported in prior literature are included in the table using the technique's name (e.g., the second row of Table B labeled 'Adv.' includes the values reported by Xie et al. (2017)). Values that we measured using our implementations of each technique are marked with asterisks. We report means and standard errors for our implementations for each method and compare to the metrics that prior methods published for each dataset.
The bottom halves of Tables 8 and 9 are separated from the top halves to indicate modified datasets. The Wass. DB baseline used the German and Adult datasets but treated protected fields differently (e.g.. by creating binary age labels at the cutoff age of 30 instead of 25, as all other techniques did). We therefore evaluated CSNs and our own implementations of Wass. DB on these different datasets as well and reported them below the horizontal line.
Interestingly, while we were able to recreate the Wass. DB results on these modified datasets, the technique, when applied to the standard datasets, demonstrated better fairness than most techniques but worse Acc.. (We repeated the hyperparameter sweeps reported by Diang et al. (2020) and used the best results.) We attribute the low Acc. to the fact that Jiang al. (020) call for linear model as opposed to deeper neural nets used by other approaches. When using the datasets suggested by Wass. DB, we reproduced their published results, did CSNs trained on the same datasets. We note, however, that predictors on this dataset are of limited use, as both Wass. DB and CSNs fail to outperform random classification accuracy.
As whole, Tables and 9 give us confidence in our implementations of the fairness baselines. Our implementations were able to match or exceed metrics reported from prior art. This suggests that our underlying implementations were correct and that new metrics we gathered on them were valid.
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Table 8: Adult dataset fairness results for CSNs and baselines. Results from our baselines are indicated with asterisks. Means (standard deviation) over 20 trials given.
Model
Acc.
s Acc.
D.I.
DD-0.5
CSN1
0.85 (0.00)
0.67 (0.00)
0.83 (0.01)
0.16 (0.01)
Adv.
0.84
0.67
Adv.*
0.85 (0.00)
0.67 (0.01)
0.87 (0.01)
0.16 (0.01)
VFAE
0.81
0.67
VFAE
0.85 (0.01)
0.70 (0.01)
0.82 (0.02)
0.17 (0.01)
FR Train
0.82
0.83
FR Train* *
0.85 (0.00)
0.67 (0.00)
0.83 (0.01)
0.16 (0.01)
Wass. DB
0.81 (0.00)
0.67 (0.00)
0.92 (0.01)
0.08 (0.01)
Random
0.76
0.67
CSN
0.76 (0.00)
0.77 (0.00)
0.98 (0.01)
0.03 (0.00)
Wass. DB
0.76
0.01
Wass. DB
0.76 (0.00)
0.77 (0.00)
1.00 (0.01) 0.01 (0.01)
Random
0.76
0.77
Table 9: German dataset fairness results for CSNs and baselines. Results from our baselines are indicated with asterisks. Means (standard deviation) over trials given.
Model
y Acc.
Acc.
D.I.
DD-0.5
CSN1
0.73 (0.02)
0.81 (0.02)
0.70 (0.12)
0.10 (0.09)
Adv.
0.74
0.81
Adv.*
0.73 (0.03)
0.81 (0.02)
0.63 (0.10)
0.10 (0.08)
VFAE
0.73
0.81
VFAE"
0.72 (0.04)
0.81 (0.04)
0.47 (0.10)
0.23 (0.09)
FR Train* *
0.72 (0.03)
0,80 (0.05)
0.55 (0.15)
0.16 (0.08)
Wass DB
0.72 (0.03)
0.81 (0.05)
0.33 (0.34)
0.02 (0.01)
Rand
0.70
0.81
CSN,
0.70 (0.03)
0.60 (0.02)
0.65 (0.13)
0.04 (0.01)
Wass. DB
0.69
0.00
Wass. DB"
0.70 (0.02)
0.60 (0.04)
0.37 (0.26)
0.02 (0.01)
Rand
0.70
0.60
18


Reviews:
[{"review_id": "jgTHq9RmVYb", "reviewer": "Reviewer_fPSK", "summary": "", "questions": "", "limitations": "", "rating": 6, "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 paper's main strengths are:\n* I found it particularly elegant to phrase both hierarchical classification and fairness in the same language, namely that of classification subspaces which are spanned by prototypes.\n* The paper connects to a wide range of concepts, namely hierarchical classification, interpretability, and fairness, such that it is of potential interest to a wide range of researchers.\n* The paper reports a wide range of experiments; so wide, indeed, that much experimental material had to be pushed to the appendix. I particularly appreciate the analysis of the hierarchies discovered by the prototype network and the comparison to the ground-truth hierarchy via edit distance.\n* The paper is clearly written. I, for one, had no problem following along and would feel well equipped to reproduce the reported results.\n\nThe paper's main weaknesses are:\n* The wide range of concepts discussed result in a certain lack of focus. Fairness, privacy, and causality are all mentioned but only discussed superficially. For fairness, this is particularly dangerous as readers may be mislead to believe that the proposed notion of orthogonality is sufficient for fairness. However, fairness has many meanings (as the paper acknowledges in the appendix) and only some of them are related to the proposed notion of orthogonality. Therefore, I would advise to revise references to fairness, privacy, and causality ind to mention explicitly that only a narrow notion of these terms is implemented by the proposed model.\n* The related work fails to mention the historic roots of the prototype concept. I understand that many recent works in prototype networks make the same mistake but I would still advise to not continue it. Prototype-based classification has - to my knowledge - been pioneered by Kohonen in the late 1980s/early 1990s with his work on Learning Vector Quantization (refer to the review by Nova and Estevez, 2014; doi: [10.1007/s00521-013-1535-3](https://doi.org/10.1007/s00521-013-1535-3) ) and has since been extended in many directions, such as metric learning (Schneider et al., 2009, doi:[10.1162/neco.2009.11-08-908](https://doi.org/10.1162/neco.2009.11-08-908)), or probabilistic classification (Seo and Obermayer, 2003, doi: [10.1162/089976603321891819](https://doi.org/10.1162/089976603321891819) ). The latter extension should be of particular interest because the classification scheme is very similar to the one proposed in this paper.\n* While the paper reports many different experiments, any single one seems relatively small with few data sets and (for hierarchical classification) few baselines. Further, I could not find information on the hyperparameter selection (e.g. how many prototypes and how strong the regularization strengths lambda were).\n\nOverall, my recommendation is to accept this paper. While the paper could be more focused, make its own contribution and limitations more clearly, and experiments could be extended, I still believe that most flaws could be addressed with minor adjustments and that the core contribution of the paper is interesting enough to a wide range of scholars that publication is warranted.\n\nNonetheless, I would appreciate if the authors could help me to deepen my understand of the work by responding to two questions:\n\n* I am not fully convinced that the projection into the plane spanned by the prototypes of a classification problem has any effect on the classification itself. If I understand correctly, the paper uses a softmax on the squared distances to all prototypes for classification (which is entirely reasonable). Now, let $D^2$ be the squared distance between a point $z$ and a prototype $p$, let $d^2$ be the squared distance between the projected point $\\tilde z$ and the same prototype $p$, and let $h^2$ be the squared distance between $z$ and $\\tilde z$. Since the distances form a right-angle triangle, we obtain $d^2 = D^2 - h^2$. This holds for any prototype in the same classification problem. Accordingly, all projected distances within one classification problem are merely the original distance minus a constant offset. This constant offset gets removed by softmax, anyways. So I would assume that the softmax probabilities are the same - no matter whether a point is projected or not.\n* Why was the parity hierarchy used as ground truth for MNIST? Garnot et al. use a hierarchy based on visual similarity of the digits (e.g. 3 and 8). Wouldn't that be more natural?", "summary_of_the_paper": "The present paper proposes a novel architecture for prototype-based classification to support class hierarchies and fairness. In particular, hierarchies are supported by training the model for multiple classification problems jointly, each in its own subspace of the feature space, spanned by the respective prototypes. For fairness, the paper proposes to make the subspace for the classification between subgroups orthogonal to all other subspaces, such that any change in subgroup membership does not influence any other classification. In a series of experiments, the paper evaluates hierarchical classification and fairness separately as well as jointly and demonstrates equal or superior results to a state-of-the-art approach from the literature.", "summary_of_the_review": "Overall, my recommendation is to accept this paper. While the paper could be more focused, make its own contribution and limitations more clearly, and experiments could be extended, I still believe that most flaws could be addressed with minor adjustments and that the core contribution of the paper is interesting enough to a wide range of scholars that publication is warranted.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "PSW6JOIhJJ7", "reviewer": "Reviewer_BhqD", "summary": "", "questions": "", "limitations": "", "rating": 3, "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 paper attempts interesting problems but lacks on 2 major fronts: (1) It is not clear what the improvement over the existing work is, and if it is significant enough to merit acceptance at ICLR (2) the writing needs a lot of work to bring out motivation for different choices.\n\nAbout the first point, the main contribution seems to lie in section 3.1, but most of the machinery here seems to be borrowed from the PCN work of Li et al. 2018. The additional contribution seems to be the concept subspace projection (if I understood correctly), whose motivation is not explained very well, and the addition of the alignment term in Eq. 1. The paper does not explain what is the additional value added by these terms over PCN.\n\nContinuing from the previous point, the paper is very hard to understand. In the second paragraph of Section 3.1, there is some departure from PCN where some combinations of $p1$ and other prototypes are taken. The process is defined in a very handwavy manner and not clear what it is the formal mathematical operation performed here. How is this different from PCN and why was this step needed? Talking about digits and concept subspaces, do we have as many concept subspaces as the number of classes? If not, how is this number picked?\n\nMoving on to the third paragraph of 3.1, the first few lines seem to be quite similar to PCN. However, at some point, a probability distribution is mentioned in connection with traditional softmax probabilities, but then yet another probability distribution is mentioned. It is not clear what the second distribution does. In the absence of formal equations, it is very difficult to understand what each component does. I would highly recommend describing each operation formally (in a sequential manner) and also adding a visualization like Figure 1 in the PCN paper to clearly convey the idea to the reader.\n\nFourth paragraph of 3.1 mentions two differences from PCN. Again, it is not clear what each of the differences achieves. Moreover, Figure 1 is neither described well in the main text nor in the caption, leaving the reader puzzled over what is happening in the figure. Instead of the regular autoencoder, a variational autoencoder is used, but again, the motivation is not clear. Other important details like the text above Equation 1, and the usage of KL diveregnce regularization term are skimmed over very quickly. The details of hierarchical classification setup in 3.4 are also glossed over quickly. The same happens in 4.2. For instance, what is meant by \"adopting the conditional probability training loss introduced by Hase et al\"?", "summary_of_the_paper": "The paper builds on prior work on prototypical classification networks (more specifically, the work of Li et al. 2018) and additionally tries to include criteria such as orthogonality to enable applications such as fair classification. An application to hierarchical networks is also described though the details are very hard to understand. Experiments show that the resulting models are able to achieve reasonable fairness accuracy tradeoffs.", "summary_of_the_review": "The contributions are not clear and the writing needs major work.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "7FV9U6_l0YI", "reviewer": "Reviewer_aHhN", "summary": "", "questions": "", "limitations": "", "rating": 3, "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": "Strength:\n- The paper proposed a new prototype-based approach considering the relationship between two concepts (classification tasks) for fairness and hierarchical classification. \n\nWeakness\n- The motivation of this paper is not clear to me. It would be helpful to understand the motivation by giving examples of major applications where both fairness and hierarchical classification should be considered. Also, is there any challenge when training a classifier using a regularization term regarding fairness in the existing hierarchy classifier training method?\n- It is not clear that why the two subspaces should be orthogonal and parallel for a fair and hierarchical classifier, respectively. Specifically, in section 3.4: \n1. For fair classification, what are the two subspaces? is it correct that the two subspaces are for label classification and sensitive attributes classification (e.g., male vs. female), respectively? Then, does the orthogonal relationship between the prototypes for estimating the sensitive attribute and label prediction guarantee the independence of the actual sensitive attribute and label prediction?\n2.  “In hierarchical classification, concepts are highly aligned and therefore parallel: the difference\nbetween a toucan and a Dalmatian is similar to the difference between a generic bird and\ndog.” --> In this example, why do parallel two concepts imply the difference between a toucan and a Dalmatian can be similar to the difference between a generic bird and dog? I think that the parallelism of two concepts is not related to the different relationships among prototypes. Then It is not clear that why two concepts should be parallel in the hierarchical classification. \n\nQuestions: \n- How does one train a hierarchical classifier with 3 or more concepts? In hierarchical classification, I think there are at least 3 concepts (e.g., dog vs bird, dog species classification, bird species classification)\n- In experiments, How was the parameter lambdas (in equation 2) chosen in the experiments?\n\nMinor feedback: \n- Adding a figure describing the proposed architecture will help readers understand the framework. \n", "summary_of_the_paper": "This paper proposed a framework (that authors called the concept subspace network) using prototype-based representation controlling the alignment between two subspaces for the purpose of the classifier (fair or hierarch classification). ", "summary_of_the_review": "The motivation of the paper and the intuition of the proposed approach is not clear. ", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "0TzXQw227Fv", "reviewer": "Reviewer_z2bK", "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": "In this paper, the ideas are quite novel and mostly well presented, and the problem handled is significant. \n\nI have though some questions and some minor comments that I hope will be addressed in the final version: \n1. The way in which concept subspaces are defined is not clear to me. In the paper, the authors write: “Given a set of k prototypes in $R^Z$ , the prototypes define a subspace generated by starting at the first prototype, $p_1$, and adding all linear combinations of vector differences from $p_1$ to all other $p_i$; $i \\in [2, k]$.” This is not clear to me, and it would be beneficial having a clearer example than the one given in the paper, in which it is not clear why we should obtain the plane $x-y$.\n2. Also, in order to get a subspace, do you need the assumption that $k < Z$? \n3. In equation (2) the term $PCN(\\cdot)$ is just defined as the loss introduced for PCN. Where is it defined?\n4. The random baseline seems to achieve very high performance in tables 1 and 2. \n5. At page 7 the authors mention a global ordering of the nodes, how was such ordering decided? \n\n\nMinor comments: \n1. $Z$ in the figure 1 instead of $z$\n2.  Add upward and downward arrows nearby the metric names to improve readability \n3. “Random” and “Rand” in Tables 1 and 2 ", "summary_of_the_paper": "The authors propose a novel model — called Concept Subspace Network (CSN) — for both hierarchical and fair classification. The idea behind the network is to use sets of prototypes to define concept subspaces in the latent space defined by the neural network itself. The relationships between the subspaces can be manipulated at training time to enforce concept relationships (i.e., two concept subspaces are orthogonal if the concepts they represent are independent, while they are parallel if the concepts they represent are hierarchically organised). ", "summary_of_the_review": "The paper can be accepted if some clarifications are made", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper extends prototypical classification networks to handle class hierarchies and fairness. New neural architecture is proposed and experimental results in support of it are presented.

Unfortunately, reviewers found that paper in its current for is not sufficiently strong to be accepted at ICLR. Authors have made a significant attempt to clarify and improve the paper in their response. However, reviewers believe that contributions and motivation can be clarified further. We encourage authors to improve their work according to the specific suggestions made by the reviewers and resubmit.
Decision: Reject