Computational Complexity for Physicists

These lecture notes are an informal introduction to the theory of computational complexity and its links to quantum computing and statistical mechanics.Comment: references updated, reprint available from http://itp.nat.uni-magdeburg.de/~mertens/papers/complexity.shtm

Manifold Elastic Net: A Unified Framework for Sparse Dimension Reduction

It is difficult to find the optimal sparse solution of a manifold learning based dimensionality reduction algorithm. The lasso or the elastic net penalized manifold learning based dimensionality reduction is not directly a lasso penalized least square problem and thus the least angle regression (LARS) (Efron et al. \cite{LARS}), one of the most popular algorithms in sparse learning, cannot be applied. Therefore, most current approaches take indirect ways or have strict settings, which can be inconvenient for applications. In this paper, we proposed the manifold elastic net or MEN for short. MEN incorporates the merits of both the manifold learning based dimensionality reduction and the sparse learning based dimensionality reduction. By using a series of equivalent transformations, we show MEN is equivalent to the lasso penalized least square problem and thus LARS is adopted to obtain the optimal sparse solution of MEN. In particular, MEN has the following advantages for subsequent classification: 1) the local geometry of samples is well preserved for low dimensional data representation, 2) both the margin maximization and the classification error minimization are considered for sparse projection calculation, 3) the projection matrix of MEN improves the parsimony in computation, 4) the elastic net penalty reduces the over-fitting problem, and 5) the projection matrix of MEN can be interpreted psychologically and physiologically. Experimental evidence on face recognition over various popular datasets suggests that MEN is superior to top level dimensionality reduction algorithms.Comment: 33 pages, 12 figure

FlashProfile: A Framework for Synthesizing Data Profiles

We address the problem of learning a syntactic profile for a collection of strings, i.e. a set of regex-like patterns that succinctly describe the syntactic variations in the strings. Real-world datasets, typically curated from multiple sources, often contain data in various syntactic formats. Thus, any data processing task is preceded by the critical step of data format identification. However, manual inspection of data to identify the different formats is infeasible in standard big-data scenarios. Prior techniques are restricted to a small set of pre-defined patterns (e.g. digits, letters, words, etc.), and provide no control over granularity of profiles. We define syntactic profiling as a problem of clustering strings based on syntactic similarity, followed by identifying patterns that succinctly describe each cluster. We present a technique for synthesizing such profiles over a given language of patterns, that also allows for interactive refinement by requesting a desired number of clusters. Using a state-of-the-art inductive synthesis framework, PROSE, we have implemented our technique as FlashProfile. Across $153$ tasks over $75$ large real datasets, we observe a median profiling time of only $\sim\,0.7\,$s. Furthermore, we show that access to syntactic profiles may allow for more accurate synthesis of programs, i.e. using fewer examples, in programming-by-example (PBE) workflows such as FlashFill.Comment: 28 pages, SPLASH (OOPSLA) 201

Beating the Perils of Non-Convexity: Guaranteed Training of Neural Networks using Tensor Methods

Training neural networks is a challenging non-convex optimization problem, and backpropagation or gradient descent can get stuck in spurious local optima. We propose a novel algorithm based on tensor decomposition for guaranteed training of two-layer neural networks. We provide risk bounds for our proposed method, with a polynomial sample complexity in the relevant parameters, such as input dimension and number of neurons. While learning arbitrary target functions is NP-hard, we provide transparent conditions on the function and the input for learnability. Our training method is based on tensor decomposition, which provably converges to the global optimum, under a set of mild non-degeneracy conditions. It consists of simple embarrassingly parallel linear and multi-linear operations, and is competitive with standard stochastic gradient descent (SGD), in terms of computational complexity. Thus, we propose a computationally efficient method with guaranteed risk bounds for training neural networks with one hidden layer.Comment: The tensor decomposition analysis is expanded, and the analysis of ridge regression is added for recovering the parameters of last layer of neural networ