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James Voss

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3 papers

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AAAI Conference 2016 Conference Paper

The Hidden Convexity of Spectral Clustering

James Voss
Mikhail Belkin
Luis Rademacher

In recent years, spectral clustering has become a standard method for data analysis used in a broad range of applications. In this paper we propose a new class of algorithms for multiway spectral clustering based on optimization of a certain “contrast function” over the unit sphere. These algorithms, partly inspired by certain Indepenent Component Analysis techniques, are simple, easy to implement and efﬁcient. Geometrically, the proposed algorithms can be interpreted as hidden basis recovery by means of function optimization. We give a complete characterization of the contrast functions admissible for provable basis recovery. We show how these conditions can be interpreted as a “hidden convexity” of our optimization problem on the sphere; interestingly, we use ef- ﬁcient convex maximization rather than the more common convex minimization. We also show encouraging experimental results on real and simulated data.

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NeurIPS Conference 2015 Conference Paper

A Pseudo-Euclidean Iteration for Optimal Recovery in Noisy ICA

James Voss
Mikhail Belkin
Luis Rademacher

Independent Component Analysis (ICA) is a popular model for blind signal separation. The ICA model assumes that a number of independent source signals are linearly mixed to form the observed signals. We propose a new algorithm, PEGI (for pseudo-Euclidean Gradient Iteration), for provable model recovery for ICA with Gaussian noise. The main technical innovation of the algorithm is to use a fixed point iteration in a pseudo-Euclidean (indefinite “inner product”) space. The use of this indefinite “inner product” resolves technical issues common to several existing algorithms for noisy ICA. This leads to an algorithm which is conceptually simple, efficient and accurate in testing. Our second contribution is combining PEGI with the analysis of objectives for optimal recovery in the noisy ICA model. It has been observed that the direct approach of demixing with the inverse of the mixing matrix is suboptimal for signal recovery in terms of the natural Signal to Interference plus Noise Ratio (SINR) criterion. There have been several partial solutions proposed in the ICA literature. It turns out that any solution to the mixing matrix reconstruction problem can be used to construct an SINR-optimal ICA demixing, despite the fact that SINR itself cannot be computed from data. That allows us to obtain a practical and provably SINR-optimal recovery method for ICA with arbitrary Gaussian noise.

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NeurIPS Conference 2013 Conference Paper

Fast Algorithms for Gaussian Noise Invariant Independent Component Analysis

James Voss
Luis Rademacher
Mikhail Belkin

The performance of standard algorithms for Independent Component Analysis quickly deteriorates under the addition of Gaussian noise. This is partially due to a common first step that typically consists of whitening, i. e. , applying Principal Component Analysis (PCA) and rescaling the components to have identity covariance, which is not invariant under Gaussian noise. In our paper we develop the first practical algorithm for Independent Component Analysis that is provably invariant under Gaussian noise. The two main contributions of this work are as follows: 1. We develop and implement a more efficient version of a Gaussian noise invariant decorrelation (quasi-orthogonalization) algorithm using Hessians of the cumulant functions. 2. We propose a very simple and efficient fixed-point GI-ICA (Gradient Iteration ICA) algorithm, which is compatible with quasi-orthogonalization, as well as with the usual PCA-based whitening in the noiseless case. The algorithm is based on a special form of gradient iteration (different from gradient descent). We provide an analysis of our algorithm demonstrating fast convergence following from the basic properties of cumulants. We also present a number of experimental comparisons with the existing methods, showing superior results on noisy data and very competitive performance in the noiseless case.

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