Local Optimality Certificates for LP Decoding of Tanner Codes

We present a new combinatorial characterization for local optimality of a codeword in an irregular Tanner code. The main novelty in this characterization is that it is based on a linear combination of subtrees in the computation trees. These subtrees may have any degree in the local code nodes and may have any height (even greater than the girth). We expect this new characterization to lead to improvements in bounds for successful decoding. We prove that local optimality in this new characterization implies ML-optimality and LP-optimality, as one would expect. Finally, we show that is possible to compute efficiently a certificate for the local optimality of a codeword given an LLR vector

Reweighted LP decoding for LDPC codes

We introduce a novel algorithm for decoding binary linear codes by linear programming. We build on the LP decoding algorithm of Feldman et al. and introduce a post-processing step that solves a second linear program that reweights the objective function based on the outcome of the original LP decoder output. Our analysis shows that for some LDPC ensembles we can improve the provable threshold guarantees compared to standard LP decoding. We also show significant empirical performance gains for the reweighted LP decoding algorithm with very small additional computational complexity

Relax, no need to round: integrality of clustering formulations

We study exact recovery conditions for convex relaxations of point cloud clustering problems, focusing on two of the most common optimization problems for unsupervised clustering: $k$-means and $k$-median clustering. Motivations for focusing on convex relaxations are: (a) they come with a certificate of optimality, and (b) they are generic tools which are relatively parameter-free, not tailored to specific assumptions over the input. More precisely, we consider the distributional setting where there are $k$ clusters in $\mathbb{R}^m$ and data from each cluster consists of $n$ points sampled from a symmetric distribution within a ball of unit radius. We ask: what is the minimal separation distance between cluster centers needed for convex relaxations to exactly recover these $k$ clusters as the optimal integral solution? For the $k$-median linear programming relaxation we show a tight bound: exact recovery is obtained given arbitrarily small pairwise separation $\epsilon > 0$ between the balls. In other words, the pairwise center separation is $\Delta > 2+\epsilon$. Under the same distributional model, the $k$-means LP relaxation fails to recover such clusters at separation as large as $\Delta = 4$. Yet, if we enforce PSD constraints on the $k$-means LP, we get exact cluster recovery at center separation $\Delta > 2\sqrt2(1+\sqrt{1/m})$. In contrast, common heuristics such as Lloyd's algorithm (a.k.a. the $k$-means algorithm) can fail to recover clusters in this setting; even with arbitrarily large cluster separation, k-means++ with overseeding by any constant factor fails with high probability at exact cluster recovery. To complement the theoretical analysis, we provide an experimental study of the recovery guarantees for these various methods, and discuss several open problems which these experiments suggest.Comment: 30 pages, ITCS 201

Reweighted LP Decoding for LDPC Codes

We introduce a novel algorithm for decoding binary linear codes by linear programming (LP). We build on the LP decoding algorithm of Feldman and introduce a postprocessing step that solves a second linear program that reweights the objective function based on the outcome of the original LP decoder output. Our analysis shows that for some LDPC ensembles we can improve the provable threshold guarantees compared to standard LP decoding. We also show significant empirical performance gains for the reweighted LP decoding algorithm with very small additional computational complexity