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Register a new userA General Computational Framework to Measure the Expressiveness of Complex Networks Using a Tighter Upper Bound of Linear Regions
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The expressiveness of deep neural network (DNN) is a perspective to understandthe surprising performance of DNN. The number of linear regions, i.e. pieces thata piece-wise-linear function represented by a DNN, is generally used to measurethe expressiveness. And the upper bound of regions number partitioned by a rec-tifier network, instead of the number itself, is a more practical measurement ofexpressiveness of a rectifier DNN. In this work, we propose a new and tighter up-per bound of regions number. Inspired by the proof of this upper bound and theframework of matrix computation in Hinz & Van de Geer (2019), we propose ageneral computational approach to compute a tight upper bound of regions numberfor theoretically any network structures (e.g. DNN with all kind of skip connec-tions and residual structures). Our experiments show our upper bound is tighterthan existing ones, and explain why skip connections and residual structures canimprove network performance.
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Graph Neural Networks (GNNs) have recently been used for node and graph classification tasks with great success, but GNNs model dependencies among the attributes of nearby neighboring nodes rather than dependencies among observed node labels. In this work, we consider the task of inductive node classification using GNNs in supervised and semi-supervised settings, with the goal of incorporating label dependencies. Because current GNNs are not universal (i.e., most-expressive) graph representations, we propose a general collective learning approach to increase the representation power of any existing GNN. Our framework combines ideas from collective classification with self-supervised learning, and uses a Monte Carlo approach to sampling embeddings for inductive learning across graphs. We evaluate performance on five real-world network datasets and demonstrate consistent, significant improvement in node classification accuracy, for a variety of state-of-the-art GNNs.
Mutual information (MI) minimization has gained considerable interests in various machine learning tasks. However, estimating and minimizing MI in high-dimensional spaces remains a challenging problem, especially when only samples, rather than distribution forms, are accessible. Previous works mainly focus on MI lower bound approximation, which is not applicable to MI minimization problems. In this paper, we propose a novel Contrastive Log-ratio Upper Bound (CLUB) of mutual information. We provide a theoretical analysis of the properties of CLUB and its variational approximation. Based on this upper bound, we introduce a MI minimization training scheme and further accelerate it with a negative sampling strategy. Simulation studies on Gaussian distributions show the reliable estimation ability of CLUB. Real-world MI minimization experiments, including domain adaptation and information bottleneck, demonstrate the effectiveness of the proposed method. The code is at https://github.com/Linear95/CLUB.
In many applications, there is a need to predict the effect of an intervention on different individuals from data. For example, which customers are persuadable by a product promotion? which patients should be treated with a certain type of treatment? These are typical causal questions involving the effect or the change in outcomes made by an intervention. The questions cannot be answered with traditional classification methods as they only use associations to predict outcomes. For personalised marketing, these questions are often answered with uplift modelling. The objective of uplift modelling is to estimate causal effect, but its literature does not discuss when the uplift represents causal effect. Causal heterogeneity modelling can solve the problem, but its assumption of unconfoundedness is untestable in data. So practitioners need guidelines in their applications when using the methods. In this paper, we use causal classification for a set of personalised decision making problems, and differentiate it from classification. We discuss the conditions when causal classification can be resolved by uplift (and causal heterogeneity) modelling methods. We also propose a general framework for causal classification, by using off-the-shelf supervised methods for flexible implementations. Experiments have shown two instantiations of the framework work for causal classification and for uplift (causal heterogeneity) modelling, and are competitive with the other uplift (causal heterogeneity) modelling methods.
In large-scale classification problems, the data set always be faced with frequent updates when a part of the data is added to or removed from the original data set. In this case, conventional incremental learning, which updates an existing classifier by explicitly modeling the data modification, is more efficient than retraining a new classifier from scratch. However, sometimes, we are more interested in determining whether we should update the classifier or performing some sensitivity analysis tasks. To deal with these such tasks, we propose an algorithm to make rational inferences about the updated linear classifier without exactly updating the classifier. Specifically, the proposed algorithm can be used to estimate the upper and lower bounds of the updated classifiers coefficient matrix with a low computational complexity related to the size of the updated dataset. Both theoretical analysis and experiment results show that the proposed approach is superior to existing methods in terms of tightness of coefficients bounds and computational complexity.
Uncertainty relations (URs) like the Heisenberg-Robertson or the time-energy UR are often considered to be hallmarks of quantum theory. Here, a simple derivation of these URs is presented based on a single classical inequality from estimation theory, a Cramer-Rao-like bound. The Heisenberg-Robertson UR is then obtained by using the Born rule and the Schrodinger equation. This allows a clear separtion of the probabilistic nature of quantum mechanics from the Hilbert space structure and the dynamical law. It also simplifies the interpretation of the bound. In addition, the Heisenberg-Robertson UR is tightened for mixed states by replacing one variance by the so-called quantum Fisher information. Thermal states of Hamiltonians with evenly-gapped energy levels are shown to saturate the tighter bound for natural choices of the operators. This example is further extended to Gaussian states of a harmonic oscillator. For many-qubit systems, we illustrate the interplay between entanglement and the structure of the operators that saturate the UR with spin-squeezed states and Dicke states.
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