Artificial neural networks achieve impressive results for various modeling tasks [LeCun et al., 2015, Wang et al., 2020]. However, a downside of their superior performance and sophisticated structure is the comprehensibility of the learned models. In many domains, it is crucial to understand the function learned by a neural network, especially when it comes to decisions that affect people [Samek et al., 2019, Molnar, 2020]. A common approach to tackle the problem of interpretability without sacrificing the superior performance is using a surrogate model as gateway to interpretability [Molnar, 2020]. Most existing global surrogate approaches use a distillation procedure to learn the surrogate model based on the predictions of the neural network [Molnar, 2020, Frosst and Hinton, 2017]. Therefore, they query the neural network based on a representative set of samples and the resulting input-output pairs are then used to train the surrogate model. This representative sample usually comprises the training data of the original model, or at least follows its distribution [Molnar, 2020, Lopes et al., 2017]. However, there are many cases where the training data cannot easily be exposed due to privacy or safety concerns [Lopes et al., 2017, Bhardwaj et al., 2019, Nayak et al., 2019].

In this paper, we use Boolean functions or low arity and low-order polynomials as examples. We present xRAI an approach for extracting symbolic However xRAI can be applied to any function family representations of the mathematical functions efficiently learnable by a neural network. For the case of loworder a neural network was supposed to learn from the polynomials, this has been shown by [Andoni et al., trained network. The approach is based on the idea 2014]. of training a so-called interpretation network that For each family of functions, we train a neural network receives the weights and biases of the trained network called interpretation network (I-Net). The I-Net receives the as input and outputs the numerical representation weights and biases of a λ-Net as input and determines an of the function the network was supposed to approximation of a target function of the trained λ-Net. We learn that can be directly translated into a symbolic train the I-Net offline by systematically training λ-Nets on representation. We show that interpretation nets for different functions from the family and using these trained different classes of functions can be trained on synthetic networks as training examples for the I-Net.

Satoshi Suzuki Hiroshi Ando ATR Human Information Processing Research Laboratories 2-2 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-02, Japan satoshi@hip.atr.co.jp, ando@hip.atr.co.jp Abstract This paper presents an unsupervised learning scheme for categorizing 3D objects from their 2D projected images. The scheme exploits an auto-associative network's ability to encode each view of a single object into a representation that indicates its view direction. We propose two models that employ different classification mechanisms; the first model selects an auto-associative network whose recovered view best matches the input view, and the second model is based on a modular architecture whose additional network classifies the views by splitting the input space nonlinearly. We demonstrate the effectiveness of the proposed classification models through simulations using 3D wire-frame objects. 1 INTRODUCTION The human visual system can recognize various 3D (three-dimensional) objects from their 2D (two-dimensional) retinal images although the images vary significantly as the viewpoint changes. Recent computational models have explored how to learn to recognize 3D objects from their projected views (Poggio & Edelman, 1990). Most existing models are, however, based on supervised learning, i.e., during training the teacher tells which object each view belongs to.

The human visual system can recognize various 3D (three-dimensional) objects from their 2D (two-dimensional) retinal images although the images vary significantly as the viewpoint changes. Recent computational models have explored how to learn to recognize 3D objects from their projected views (Poggio & Edelman, 1990). Most existing models are, however, based on supervised learning, i.e., during training the teacher tells which object each view belongs to. The model proposed by Weinshall et al. (1990) also requires a signal that segregates different objects during training. This paper, on the other hand, discusses unsupervised aspects of 3D object recognition where the system discovers categories by itself.

The human visual system can recognize various 3D (three-dimensional) objects from their 2D (two-dimensional) retinal images although the images vary significantly as the viewpoint changes. Recent computational models have explored how to learn to recognize 3D objects from their projected views (Poggio & Edelman, 1990). Most existing models are, however, based on supervised learning, i.e., during training the teacher tells which object each view belongs to. The model proposed by Weinshall et al. (1990) also requires a signal that segregates different objects during training. This paper, on the other hand, discusses unsupervised aspects of 3D object recognition where the system discovers categories by itself.