Most existing interpretable methods explain a black-box model in a post-hoc manner, which uses simpler models or data analysis techniques to interpret the predictions after the model is learned. However, they (a) may derive contradictory explanations on the same predictions given different methods and data samples, and (b) focus on using simpler models to provide higher descriptive accuracy at the sacrifice of prediction accuracy. To address these issues, we propose a hybrid interpretable model that combines a piecewise linear component and a nonlinear component. The first component describes the explicit feature contributions by piecewise linear approximation to increase the expressiveness of the model. The other component uses a multi-layer perceptron to capture feature interactions and implicit nonlinearity, and increase the prediction performance. Different from the post-hoc approaches, the interpretability is obtained once the model is learned in the form of feature shapes. We also provide a variant to explore higher-order interactions among features to demonstrate that the proposed model is flexible for adaptation. Experiments demonstrate that the proposed model can achieve good interpretability by describing feature shapes while maintaining state-of-the-art accuracy.
Today, it is more important than ever before for users to have trust in the models they use. As Machine Learning models fall under increased regulatory scrutiny and begin to see more applications in high-stakes situations, it becomes critical to explain our models. Piecewise Linear Neural Networks (PLNN) with the ReLU activation function have quickly become extremely popular models due to many appealing properties; however, they still present many challenges in the areas of robustness and interpretation. To this end, we introduce novel methodology toward simplification and increased interpreta…
This paper proposes two bottom-up interpretable neural network (NN) constructions for universal approximation, namely Triangularly-constructed NN (TNN) and Semi-Quantized Activation NN (SQANN). Further notable properties are (1) resistance to catastrophic forgetting (2) existence of proof for arbitrarily high accuracies (3) the ability to identify samples that are out-of-distribution through interpretable activation "fingerprints".
Verification of Geometric Robustness of Neural Networks via Piecewise Linear Approximation and Lipschitz Optimisation
Imperial College London（帝国理工学院）；University of California San Diego（加利福尼亚大学圣迭戈分校）；Inria（法国国家信息与自动化研究所）；École Normale Supérieure（巴黎高等师范学院）；PSL University（巴黎文理研究大学）；CNRS（法国国家科学研究中心）；University of Limassol（利马索尔大学）；Safe Intelligence
Deep learning models have achieved state-of-the-art performance in many classification tasks. However, most of them cannot provide an interpretation for their classification results. Machine learning models that are interpretable are usually linear or piecewise linear and yield inferior performance. Non-linear models achieve much better classification performance, but it is hard to interpret their classification results. This may have been changed by an interpretable feedforward neural network (IFFNN) proposed that achieves both high classification performance and interpretability for malware …
KJ Somaiya College of Engineering（KJ Somaiya工程学院）；National Institute of Technology Karnataka, Surathkal（卡纳塔克国立理工学院苏拉特卡尔分校）
Relu Fully Connected Networks are ubiquitous but uninterpretable because they fit piecewise linear functions emerging from multi-layered structures and complex interactions of model weights. This paper takes a novel approach to piecewise fits by using set operations on individual pieces(parts). This is done by approximating canonical normal forms and using the resultant as a model. This gives special advantages like (a)strong correspondence of parameters to pieces of the fit function(High Interpretability); (b)ability to fit any combination of continuous functions as pieces of the piecewise fu…
CLARITree: Cholesky and Lookahead Accelerations for Regression with Interpretable Piecewise Linear Trees
Department of Computer Science, Duke University, Durham, USA（杜克大学计算机科学系，美国达勒姆）；Department of Computer Science, University of British Columbia, Vancouver, Canada（不列颠哥伦比亚大学计算机科学系，加拿大温哥华）
Existing works on "black-box" model interpretation use local-linear approximations to explain the predictions made for each data instance in terms of the importance assigned to the different features for arriving at the prediction. These works provide instancewise explanations and thus give a local view of the model. To be able to trust the model it is important to understand the global model behavior and there are relatively fewer works which do the same. Piecewise local-linear models provide a natural way to extend local-linear models to explain the global behavior of the model. In this work…
University of Tsukuba（筑波大学）；Senshu University（专修大学）；Tokyo University of Agriculture and Technology（东京农工大学）
This paper concerns a method of selecting a subset of features for a sequential logit model. Tanaka and Nakagawa (2014) proposed a mixed integer quadratic optimization formulation for solving the problem based on a quadratic approximation of the logistic loss function. However, since there is a significant gap between the logistic loss function and its quadratic approximation, their formulation may fail to find a good subset of features. To overcome this drawback, we apply a piecewise-linear approximation to the logistic loss function. Accordingly, we frame the feature subset selection problem…
University of Southern California（南加利福尼亚大学）；Army Research Laboratory（陆军研究实验室）
The construction of a multilayer perceptron (MLP) as a piecewise low-order polynomial approximator using a signal processing approach is presented in this work. The constructed MLP contains one input, one intermediate and one output layers. Its construction includes the specification of neuron numbers and all filter weights. Through the construction, a one-to-one correspondence between the approximation of an MLP and that of a piecewise low-order polynomial is established. Comparison between piecewise polynomial and MLP approximations is made. Since the approximation capability of piecewise lo…
Energy Optimized Piecewise Polynomial Approximation Utilizing Modern Machine Learning Optimizers
Josef Ressel Centre for Intelligent and Secure Industrial Automation（智能与安全工业自动化约瑟夫·雷斯尔中心）；Salzburg University of Applied Sciences（萨尔茨堡应用科技大学）；Paris Lodron University Salzburg（萨尔茨堡巴黎洛登大学）
National Technical University of Athens（雅典国家技术大学）；Harvard University（哈佛大学）
Tropical Geometry and Mathematical Morphology share the same max-plus and min-plus semiring arithmetic and matrix algebra. In this chapter we summarize some of their main ideas and common (geometric and algebraic) structure, generalize and extend both of them using weighted lattices and a max-$\star$ algebra with an arbitrary binary operation $\star$ that distributes over max, and outline applications to geometry, machine learning, and optimization. Further, we generalize tropical geometrical objects using weighted lattices. Finally, we provide the optimal solution of max-$\star$ equations usi…
We present an explicit construction for feedforward neural network (FNN), which provides a piecewise constant approximation for multivariate functions. The proposed FNN has two hidden layers, where the weights and thresholds are explicitly defined and do not require numerical optimization for training. Unlike most of the existing work on explicit FNN construction, the proposed FNN does not rely on tensor structure in multiple dimensions. Instead, it automatically creates Voronoi tessellation of the domain, based on the given data of the target function, and piecewise constant approximation of …
We study the necessary and sufficient complexity of ReLU neural networks---in terms of depth and number of weights---which is required for approximating classifier functions in $L^2$. As a model class, we consider the set $\mathcal{E}^β(\mathbb R^d)$ of possibly discontinuous piecewise $C^β$ functions $f : [-1/2, 1/2]^d \to \mathbb R$, where the different smooth regions of $f$ are separated by $C^β$ hypersurfaces. For dimension $d \geq 2$, regularity $β> 0$, and accuracy $\varepsilon > 0$, we construct artificial neural networks with ReLU activation function that approximate functions from $\m…
Harbin Institute of Technology, Shenzhen（哈尔滨工业大学（深圳））；Delft University of Technology（代尔夫特理工大学）；Shanghai Jiao Tong University（上海交通大学）
In this paper, the disjunctive and conjunctive lattice piecewise affine (PWA) approximations of explicit linear model predictive control (MPC) are proposed. The training data are generated uniformly in the domain of interest, consisting of the state samples and corresponding affine control laws, based on which the lattice PWA approximations are constructed. Re-sampling of data is also proposed to guarantee that the lattice PWA approximations are identical to explicit MPC control law in the unique order (UO) regions containing the sample points as interior points. Additionally, under mild assum…
UC Berkeley（加州大学伯克利分校）；University of Edinburgh（爱丁堡大学）；Columbia University（哥伦比亚大学）
We give a highly efficient "semi-agnostic" algorithm for learning univariate probability distributions that are well approximated by piecewise polynomial density functions. Let $p$ be an arbitrary distribution over an interval $I$ which is $τ$-close (in total variation distance) to an unknown probability distribution $q$ that is defined by an unknown partition of $I$ into $t$ intervals and $t$ unknown degree-$d$ polynomials specifying $q$ over each of the intervals. We give an algorithm that draws $\tilde{O}(t\new{(d+1)}/\eps^2)$ samples from $p$, runs in time $\poly(t,d,1/\eps)$, and with hig…
University of Debrecen（德布勒森大学）；Analytical Minds Ltd.（分析思维有限公司）
The recently introduced Matching by Tone Mapping (MTM) dissimilarity measure enables template matching under smooth non-linear distortions and also has a well-established mathematical background. MTM operates by binning the template, but the ideal binning for a particular problem is an open question. By pointing out an important analogy between the well known mutual information (MI) and MTM, we introduce the term "normalized unexplained variance" (nUV) for MTM to emphasize its relevance and applicability beyond image processing. Then, we provide theoretical results on the optimal binning techn…
Nonlinear Systems Laboratory Massachusetts Institute of Technology（非线性系统实验室 麻省理工学院）
We discuss technical results on learning function approximations using piecewise-linear basis functions, and analyze their stability and convergence using nonlinear contraction theory.
Recently, the interpretability of deep learning has attracted a lot of attention. A plethora of methods have attempted to explain neural networks by feature visualization, saliency maps, model distillation, and so on. However, it is hard for these methods to reveal the intrinsic properties of neural networks. In this work, we studied the 1-D optimal piecewise linear approximation (PWLA) problem, and associated it with a designed neural network, named lattice neural network (LNN). We asked four essential questions as following: (1) What are the characters of the optimal solution of the PWLA pro…
University of Basel（巴塞尔大学）；University of Zürich（苏黎世大学）
Most stochastic gradient descent algorithms can optimize neural networks that are sub-differentiable in their parameters; however, this implies that the neural network's activation function must exhibit a degree of continuity which limits the neural network model's uniform approximation capacity to continuous functions. This paper focuses on the case where the discontinuities arise from distinct sub-patterns, each defined on different parts of the input space. We propose a new discontinuous deep neural network model trainable via a decoupled two-step procedure that avoids passing gradient upda…
In this paper we apply the previously introduced approximation method based on the ANOVA (analysis of variance) decomposition and Grouped Transformations to synthetic and real data. The advantage of this method is the interpretability of the approximation, i.e., the ability to rank the importance of the attribute interactions or the variable couplings. Moreover, we are able to generate an attribute ranking to identify unimportant variables and reduce the dimensionality of the problem. We compare the method to other approaches on publicly available benchmark datasets.
