Distributed (Δ+1)(\Delta+1)-Coloring in Sublogarithmic Rounds

We give a new randomized distributed algorithm for $(\Delta+1)$-coloring in the LOCAL model, running in $O(\sqrt{\log \Delta})+ 2^{O(\sqrt{\log \log n})}$ rounds in a graph of maximum degree~$\Delta$. This implies that the $(\Delta+1)$-coloring problem is easier than the maximal independent set problem and the maximal matching problem, due to their lower bounds of $\Omega \left( \min \left( \sqrt{\frac{\log n}{\log \log n}}, \frac{\log \Delta}{\log \log \Delta} \right) \right)$ by Kuhn, Moscibroda, and Wattenhofer [PODC'04]. Our algorithm also extends to list-coloring where the palette of each node contains $\Delta+1$ colors. We extend the set of distributed symmetry-breaking techniques by performing a decomposition of graphs into dense and sparse parts

Round Compression for Parallel Matching Algorithms

For over a decade now we have been witnessing the success of {\em massive parallel computation} (MPC) frameworks, such as MapReduce, Hadoop, Dryad, or Spark. One of the reasons for their success is the fact that these frameworks are able to accurately capture the nature of large-scale computation. In particular, compared to the classic distributed algorithms or PRAM models, these frameworks allow for much more local computation. The fundamental question that arises in this context is though: can we leverage this additional power to obtain even faster parallel algorithms? A prominent example here is the {\em maximum matching} problem---one of the most classic graph problems. It is well known that in the PRAM model one can compute a 2-approximate maximum matching in $O(\log{n})$ rounds. However, the exact complexity of this problem in the MPC framework is still far from understood. Lattanzi et al. showed that if each machine has $n^{1+\Omega(1)}$ memory, this problem can also be solved $2$-approximately in a constant number of rounds. These techniques, as well as the approaches developed in the follow up work, seem though to get stuck in a fundamental way at roughly $O(\log{n})$ rounds once we enter the near-linear memory regime. It is thus entirely possible that in this regime, which captures in particular the case of sparse graph computations, the best MPC round complexity matches what one can already get in the PRAM model, without the need to take advantage of the extra local computation power. In this paper, we finally refute that perplexing possibility. That is, we break the above $O(\log n)$ round complexity bound even in the case of {\em slightly sublinear} memory per machine. In fact, our improvement here is {\em almost exponential}: we are able to deliver a $(2+\epsilon)$-approximation to maximum matching, for any fixed constant $\epsilon>0$, in $O((\log \log n)^2)$ rounds

Do algorithms and barriers for sparse principal component analysis extend to other structured settings?

We study a principal component analysis problem under the spiked Wishart model in which the structure in the signal is captured by a class of union-of-subspace models. This general class includes vanilla sparse PCA as well as its variants with graph sparsity. With the goal of studying these problems under a unified statistical and computational lens, we establish fundamental limits that depend on the geometry of the problem instance, and show that a natural projected power method exhibits local convergence to the statistically near-optimal neighborhood of the solution. We complement these results with end-to-end analyses of two important special cases given by path and tree sparsity in a general basis, showing initialization methods and matching evidence of computational hardness. Overall, our results indicate that several of the phenomena observed for vanilla sparse PCA extend in a natural fashion to its structured counterparts

Matching Image Sets via Adaptive Multi Convex Hull

Traditional nearest points methods use all the samples in an image set to construct a single convex or affine hull model for classification. However, strong artificial features and noisy data may be generated from combinations of training samples when significant intra-class variations and/or noise occur in the image set. Existing multi-model approaches extract local models by clustering each image set individually only once, with fixed clusters used for matching with various image sets. This may not be optimal for discrimination, as undesirable environmental conditions (eg. illumination and pose variations) may result in the two closest clusters representing different characteristics of an object (eg. frontal face being compared to non-frontal face). To address the above problem, we propose a novel approach to enhance nearest points based methods by integrating affine/convex hull classification with an adapted multi-model approach. We first extract multiple local convex hulls from a query image set via maximum margin clustering to diminish the artificial variations and constrain the noise in local convex hulls. We then propose adaptive reference clustering (ARC) to constrain the clustering of each gallery image set by forcing the clusters to have resemblance to the clusters in the query image set. By applying ARC, noisy clusters in the query set can be discarded. Experiments on Honda, MoBo and ETH-80 datasets show that the proposed method outperforms single model approaches and other recent techniques, such as Sparse Approximated Nearest Points, Mutual Subspace Method and Manifold Discriminant Analysis.Comment: IEEE Winter Conference on Applications of Computer Vision (WACV), 201

Personalized Dictionary Learning for Heterogeneous Datasets

We introduce a relevant yet challenging problem named Personalized Dictionary Learning (PerDL), where the goal is to learn sparse linear representations from heterogeneous datasets that share some commonality. In PerDL, we model each dataset's shared and unique features as global and local dictionaries. Challenges for PerDL not only are inherited from classical dictionary learning (DL), but also arise due to the unknown nature of the shared and unique features. In this paper, we rigorously formulate this problem and provide conditions under which the global and local dictionaries can be provably disentangled. Under these conditions, we provide a meta-algorithm called Personalized Matching and Averaging (PerMA) that can recover both global and local dictionaries from heterogeneous datasets. PerMA is highly efficient; it converges to the ground truth at a linear rate under suitable conditions. Moreover, it automatically borrows strength from strong learners to improve the prediction of weak learners. As a general framework for extracting global and local dictionaries, we show the application of PerDL in different learning tasks, such as training with imbalanced datasets and video surveillance