247,170 research outputs found
Sample Complexity Analysis for Learning Overcomplete Latent Variable Models through Tensor Methods
We provide guarantees for learning latent variable models emphasizing on the
overcomplete regime, where the dimensionality of the latent space can exceed
the observed dimensionality. In particular, we consider multiview mixtures,
spherical Gaussian mixtures, ICA, and sparse coding models. We provide tight
concentration bounds for empirical moments through novel covering arguments. We
analyze parameter recovery through a simple tensor power update algorithm. In
the semi-supervised setting, we exploit the label or prior information to get a
rough estimate of the model parameters, and then refine it using the tensor
method on unlabeled samples. We establish that learning is possible when the
number of components scales as , where is the observed
dimension, and is the order of the observed moment employed in the tensor
method. Our concentration bound analysis also leads to minimax sample
complexity for semi-supervised learning of spherical Gaussian mixtures. In the
unsupervised setting, we use a simple initialization algorithm based on SVD of
the tensor slices, and provide guarantees under the stricter condition that
(where constant can be larger than ), where the
tensor method recovers the components under a polynomial running time (and
exponential in ). Our analysis establishes that a wide range of
overcomplete latent variable models can be learned efficiently with low
computational and sample complexity through tensor decomposition methods.Comment: Title change
Finding Disjoint Paths on Edge-Colored Graphs: More Tractability Results
The problem of finding the maximum number of vertex-disjoint uni-color paths
in an edge-colored graph (called MaxCDP) has been recently introduced in
literature, motivated by applications in social network analysis. In this paper
we investigate how the complexity of the problem depends on graph parameters
(namely the number of vertices to remove to make the graph a collection of
disjoint paths and the size of the vertex cover of the graph), which makes
sense since graphs in social networks are not random and have structure. The
problem was known to be hard to approximate in polynomial time and not
fixed-parameter tractable (FPT) for the natural parameter. Here, we show that
it is still hard to approximate, even in FPT-time. Finally, we introduce a new
variant of the problem, called MaxCDDP, whose goal is to find the maximum
number of vertex-disjoint and color-disjoint uni-color paths. We extend some of
the results of MaxCDP to this new variant, and we prove that unlike MaxCDP,
MaxCDDP is already hard on graphs at distance two from disjoint paths.Comment: Journal version in JOC
- …