A variant of the Johnson-Lindenstrauss lemma for circulant matrices

We continue our study of the Johnson-Lindenstrauss lemma and its connection to circulant matrices started in \cite{HV}. We reduce the bound on $k$ from $k=O(\epsilon^{-2}\log^3n)$ proven there to $k=O(\epsilon^{-2}\log^2n)$. Our technique differs essentially from the one used in \cite{HV}. We employ the discrete Fourier transform and singular value decomposition to deal with the dependency caused by the circulant structure

A Latent Source Model for Patch-Based Image Segmentation

Despite the popularity and empirical success of patch-based nearest-neighbor and weighted majority voting approaches to medical image segmentation, there has been no theoretical development on when, why, and how well these nonparametric methods work. We bridge this gap by providing a theoretical performance guarantee for nearest-neighbor and weighted majority voting segmentation under a new probabilistic model for patch-based image segmentation. Our analysis relies on a new local property for how similar nearby patches are, and fuses existing lines of work on modeling natural imagery patches and theory for nonparametric classification. We use the model to derive a new patch-based segmentation algorithm that iterates between inferring local label patches and merging these local segmentations to produce a globally consistent image segmentation. Many existing patch-based algorithms arise as special cases of the new algorithm.Comment: International Conference on Medical Image Computing and Computer Assisted Interventions 201

Scalable and Robust Community Detection with Randomized Sketching

This paper explores and analyzes the unsupervised clustering of large partially observed graphs. We propose a scalable and provable randomized framework for clustering graphs generated from the stochastic block model. The clustering is first applied to a sub-matrix of the graph's adjacency matrix associated with a reduced graph sketch constructed using random sampling. Then, the clusters of the full graph are inferred based on the clusters extracted from the sketch using a correlation-based retrieval step. Uniform random node sampling is shown to improve the computational complexity over clustering of the full graph when the cluster sizes are balanced. A new random degree-based node sampling algorithm is presented which significantly improves upon the performance of the clustering algorithm even when clusters are unbalanced. This algorithm improves the phase transitions for matrix-decomposition-based clustering with regard to computational complexity and minimum cluster size, which are shown to be nearly dimension-free in the low inter-cluster connectivity regime. A third sampling technique is shown to improve balance by randomly sampling nodes based on spatial distribution. We provide analysis and numerical results using a convex clustering algorithm based on matrix completion

Almost Optimal Unrestricted Fast Johnson-Lindenstrauss Transform

The problems of random projections and sparse reconstruction have much in common and individually received much attention. Surprisingly, until now they progressed in parallel and remained mostly separate. Here, we employ new tools from probability in Banach spaces that were successfully used in the context of sparse reconstruction to advance on an open problem in random pojection. In particular, we generalize and use an intricate result by Rudelson and Vershynin for sparse reconstruction which uses Dudley's theorem for bounding Gaussian processes. Our main result states that any set of $N = \exp(\tilde{O}(n))$ real vectors in $n$ dimensional space can be linearly mapped to a space of dimension k=O(\log N\polylog(n)), while (1) preserving the pairwise distances among the vectors to within any constant distortion and (2) being able to apply the transformation in time $O(n\log n)$ on each vector. This improves on the best known $N = \exp(\tilde{O}(n^{1/2}))$ achieved by Ailon and Liberty and $N = \exp(\tilde{O}(n^{1/3}))$ by Ailon and Chazelle. The dependence in the distortion constant however is believed to be suboptimal and subject to further investigation. For constant distortion, this settles the open question posed by these authors up to a \polylog(n) factor while considerably simplifying their constructions