Nonparametric Nearest Neighbor Random Process Clustering

We consider the problem of clustering noisy finite-length observations of stationary ergodic random processes according to their nonparametric generative models without prior knowledge of the model statistics and the number of generative models. Two algorithms, both using the L1-distance between estimated power spectral densities (PSDs) as a measure of dissimilarity, are analyzed. The first algorithm, termed nearest neighbor process clustering (NNPC), to the best of our knowledge, is new and relies on partitioning the nearest neighbor graph of the observations via spectral clustering. The second algorithm, simply referred to as k-means (KM), consists of a single k-means iteration with farthest point initialization and was considered before in the literature, albeit with a different measure of dissimilarity and with asymptotic performance results only. We show that both NNPC and KM succeed with high probability under noise and even when the generative process PSDs overlap significantly, all provided that the observation length is sufficiently large. Our results quantify the tradeoff between the overlap of the generative process PSDs, the noise variance, and the observation length. Finally, we present numerical performance results for synthetic and real data.Comment: IEEE International Symposium on Information Theory (ISIT), June 2015, to appea

Subspace clustering of dimensionality-reduced data

Subspace clustering refers to the problem of clustering unlabeled high-dimensional data points into a union of low-dimensional linear subspaces, assumed unknown. In practice one may have access to dimensionality-reduced observations of the data only, resulting, e.g., from "undersampling" due to complexity and speed constraints on the acquisition device. More pertinently, even if one has access to the high-dimensional data set it is often desirable to first project the data points into a lower-dimensional space and to perform the clustering task there; this reduces storage requirements and computational cost. The purpose of this paper is to quantify the impact of dimensionality-reduction through random projection on the performance of the sparse subspace clustering (SSC) and the thresholding based subspace clustering (TSC) algorithms. We find that for both algorithms dimensionality reduction down to the order of the subspace dimensions is possible without incurring significant performance degradation. The mathematical engine behind our theorems is a result quantifying how the affinities between subspaces change under random dimensionality reducing projections.Comment: ISIT 201

The AutoProof Verifier: Usability by Non-Experts and on Standard Code

Formal verification tools are often developed by experts for experts; as a result, their usability by programmers with little formal methods experience may be severely limited. In this paper, we discuss this general phenomenon with reference to AutoProof: a tool that can verify the full functional correctness of object-oriented software. In particular, we present our experiences of using AutoProof in two contrasting contexts representative of non-expert usage. First, we discuss its usability by students in a graduate course on software verification, who were tasked with verifying implementations of various sorting algorithms. Second, we evaluate its usability in verifying code developed for programming assignments of an undergraduate course. The first scenario represents usability by serious non-experts; the second represents usability on "standard code", developed without full functional verification in mind. We report our experiences and lessons learnt, from which we derive some general suggestions for furthering the development of verification tools with respect to improving their usability.Comment: In Proceedings F-IDE 2015, arXiv:1508.0338

Greedy Algorithms for Cone Constrained Optimization with Convergence Guarantees

Greedy optimization methods such as Matching Pursuit (MP) and Frank-Wolfe (FW) algorithms regained popularity in recent years due to their simplicity, effectiveness and theoretical guarantees. MP and FW address optimization over the linear span and the convex hull of a set of atoms, respectively. In this paper, we consider the intermediate case of optimization over the convex cone, parametrized as the conic hull of a generic atom set, leading to the first principled definitions of non-negative MP algorithms for which we give explicit convergence rates and demonstrate excellent empirical performance. In particular, we derive sublinear ($\mathcal{O}(1/t)$) convergence on general smooth and convex objectives, and linear convergence ($\mathcal{O}(e^{-t})$) on strongly convex objectives, in both cases for general sets of atoms. Furthermore, we establish a clear correspondence of our algorithms to known algorithms from the MP and FW literature. Our novel algorithms and analyses target general atom sets and general objective functions, and hence are directly applicable to a large variety of learning settings.Comment: NIPS 201