The question of what makes a data distribution suitable for deep learning is a fundamental open problem. Focusing on locally connected neural networks (a prevalent family of architectures that includes convolutional and recurrent neural networks as well as local self-attention models), we address this problem by adopting theoretical tools from quantum physics. Our main theoretical result states that a certain locally connected neural network is capable of accurate prediction over a data distribution if and only if the data distribution admits low quantum entanglement under certain canonical partitions of features. As a practical application of this result, we derive a preprocessing method for enhancing the suitability of a data distribution to locally connected neural networks. Experiments with widespread models over various datasets demonstrate our findings. We hope that our use of quantum entanglement will encourage further adoption of tools from physics for formally reasoning about the relation between deep learning and real-world data.

Global optimization of decision trees has shown to be promising in terms of accuracy, size, and consequently human comprehensibility. However, many of the methods used rely on general-purpose solvers for which scalability remains an issue.Dynamic programming methods have been shown to scale much better because they exploit the tree structure by solving subtrees as independent subproblems. However, this only works when an objective can be optimized separately for subtrees.We explore this relationship in detail and show the necessary and sufficient conditions for such separability and generalize previous dynamic programming approaches into a framework that can optimize any combination of separable objectives and constraints.Experiments on five application domains show the general applicability of this framework, while outperforming the scalability of general-purpose solvers by a large margin.

The question of what makes a data distribution suitable for deep learning is a fundamental open problem. Focusing on locally connected neural networks (a prevalent family of architectures that includes convolutional and recurrent neural networks as well as local self-attention models), we address this problem by adopting theoretical tools from quantum physics. Our main theoretical result states that a certain locally connected neural network is capable of accurate prediction over a data distribution if and only if the data distribution admits low quantum entanglement under certain canonical partitions of features. As a practical application of this result, we derive a preprocessing method for enhancing the suitability of a data distribution to locally connected neural networks. Experiments with widespread models over various datasets demonstrate our findings.

Global optimization of decision trees has shown to be promising in terms of accuracy, size, and consequently human comprehensibility. However, many of the methods used rely on general-purpose solvers for which scalability remains an issue.Dynamic programming methods have been shown to scale much better because they exploit the tree structure by solving subtrees as independent subproblems. However, this only works when an objective can be optimized separately for subtrees.We explore this relationship in detail and show the necessary and sufficient conditions for such separability and generalize previous dynamic programming approaches into a framework that can optimize any combination of separable objectives and constraints.Experiments on five application domains show the general applicability of this framework, while outperforming the scalability of general-purpose solvers by a large margin.

We study the identification of direct and indirect causes on time series and provide conditions in the presence of latent variables, which we prove to be necessary and sufficient under some graph constraints. Our theoretical results and estimation algorithms require two conditional independence tests for each observed candidate time series to determine whether or not it is a cause of an observed target time series. We provide experimental results in simulations, as well as real data. Our results show that our method leads to very low false positives and relatively low false negative rates, outperforming the widely used Granger causality.

The simplicial condition and other stronger conditions that imply it have recently played a central role in developing polynomial time algorithms with provable asymptotic consistency and sample complexity guarantees for topic estimation in separable topic models. Of these algorithms, those that rely solely on the simplicial condition are impractical while the practical ones need stronger conditions. In this paper, we demonstrate, for the first time, that the simplicial condition is a fundamental, algorithm-independent, information-theoretic necessary condition for consistent separable topic estimation. Furthermore, under solely the simplicial condition, we present a practical quadratic-complexity algorithm based on random projections which consistently detects all novel words of all topics using only up to second-order empirical word moments. This algorithm is amenable to distributed implementation making it attractive for "big-data" scenarios involving a network of large distributed databases.

This paper formulates a necessary and sufficient condition for a generic graph matching problem to be equivalent to the maximum vertex and edge weight clique problem in a derived association graph. The consequences of this results are threefold: first, the condition is general enough to cover a broad range of practical graph matching problems; second, a proof to establish equivalence between graph matching and clique search reduces to showing that a given graph matching problem satisfies the proposed condition; and third, the result sets the scene for generic continuous solutions for a broad range of graph matching problems. To illustrate the mathematical framework, we apply it to a number of graph matching problems, including the problem of determining the graph edit distance.