We introduce the diffusion @math -means clustering method on Riemannian submanifolds, which maximizes the within-cluster connectedness based on the diffusion distance. The diffusion @math -means constructs a random walk on the similarity graph with vertices as data points randomly sampled on the manifolds and edges as similarities given by a kernel that captures the local geometry of manifolds. Thus the diffusion @math -means is a multi-scale clustering tool that is suitable for data with non-linear and non-Euclidean geometric features in mixed dimensions. Given the number of clusters, we propose a polynomial-time convex relaxation algorithm via the semidefinite programming (SDP) to solve the diffusion @math -means. In addition, we also propose a nuclear norm (i.e., trace norm) regularized SDP that is adaptive to the number of clusters. In both cases, we show that exact recovery of the SDPs for diffusion @math -means can be achieved under suitable between-cluster separability and within-cluster connectedness of the submanifolds, which together quantify the hardness of the manifold clustering problem. We further propose the localized diffusion @math -means by using the local adaptive bandwidth estimated from the nearest neighbors. We show that exact recovery of the localized diffusion @math -means is fully adaptive to the local probability density and geometric structures of the underlying submanifolds.
On the contrary, literature on theoretical guarantees for manifold clustering is rather scarce, with the exception @cite_6 . Near-optimal exact recovery of some emblematic clustering methods based on pairwise distances of data is derived under a condition that the minimal signal separation strength over all pairs of submanifolds is larger than a threshold. Compared with our diffusion @math -means with local scaling, results established in @cite_6 are non-adaptive to the local density and (geometric) structures of the submanifolds (cf. Theorem and ahead).
@cite_6
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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: @math -means and @math -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 @math clusters in @math and data from each cluster consists of @math 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 @math clusters as the optimal integral solution? For the @math -median linear programming relaxation we show a tight bound: exact recovery is obtained given arbitrarily small pairwise separation @math between the balls. In other words, the pairwise center separation is @math . Under the same distributional model, the @math -means LP relaxation fails to recover such clusters at separation as large as @math . Yet, if we enforce PSD constraints on the @math -means LP, we get exact cluster recovery at center separation @math . In contrast, common heuristics such as Lloyd's algorithm (a.k.a. the @math -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. This paper advocates a novel framework for segmenting a dataset in a Riemannian manifold @math into clusters lying around low-dimensional submanifolds of @math . Important examples of @math , for which the proposed clustering algorithm is computationally efficient, are the sphere, the set of positive definite matrices, and the Grassmannian. The clustering problem with these examples of @math is already useful for numerous application domains such as action identification in video sequences, dynamic texture clustering, brain fiber segmentation in medical imaging, and clustering of deformed images. The proposed clustering algorithm constructs a data-affinity matrix by thoroughly exploiting the intrinsic geometry and then applies spectral clustering. The intrinsic local geometry is encoded by local sparse coding and more importantly by directional information of local tangent spaces and geodesics. Theoretical guarantees are established for a simplified variant of the algorithm even when the clusters intersect. To avoid complication, these guarantees assume that the underlying submanifolds are geodesic. Extensive validation on synthetic and real data demonstrates the resiliency of the proposed method against deviations from the theoretical model as well as its superior performance over state-of-the-art techniques. In the context of clustering, we consider a generative model in a Euclidean ambient space with clusters of different shapes, dimensions, sizes, and densities. In an asymptotic setting where the number of points becomes large, we obtain theoretical guaranties for some emblematic methods based on pairwise distances: a simple algorithm based on the extraction of connected components in a neighborhood graph; hierarchical clustering with single linkage; and the spectral clustering method of Ng, Jordan, and Weiss. The methods are shown to enjoy some near-optimal properties in terms of separation between clusters and robustness to outliers. The local scaling method of Zelnik-Manor and Perona is shown to lead to a near-optimal choice for the scale in the first and third methods. We also provide a lower bound on the spectral gap to consistently choose the correct number of clusters in the spectral method. Let @math be a geometrically bounded symplectic manifold, @math be a closed, regular coisotropic submanifold, and @math be a Hamiltonian diffeomorphism. The main result of this article is that the number of leafwise fixed points of @math is bounded below by the sum of the Betti numbers of @math , provided that the Hofer distance between @math and the identity is small enough and the pair @math is non-degenerate. As an application, I prove a presymplectic non-embedding result. A version of the Arnold-Givental conjecture for coisotropic submanifolds is also discussed. We propose a novel algorithm for clustering data sampled from multiple submanifolds of a Riemannian manifold. First, we learn a representation of the data using generalizations of local nonlinear dimensionality reduction algorithms from Euclidean to Riemannian spaces. Such generalizations exploit geometric properties of the Riemannian space, particularly its Riemannian metric. Then, assuming that the data points from different groups are separated, we show that the null space of a matrix built from the local representation gives the segmentation of the data. Our method is computationally simple and performs automatic segmentation without requiring user initialization. We present results on 2-D motion segmentation and diffusion tensor imaging segmentation. We consider the problem of estimating the discrete clustering structures under Sub-Gaussian Mixture Models. Our main results establish a hidden integrality property of a semidefinite programming (SDP) relaxation for this problem: while the optimal solutions to the SDP are not integer-valued in general, their estimation errors can be upper bounded in terms of the error of an idealized integer program. The error of the integer program, and hence that of the SDP, are further shown to decay exponentially in the signal-to-noise ratio. To the best of our knowledge, this is the first exponentially decaying error bound for convex relaxations of mixture models, and our results reveal the "global-to-local" mechanism that drives the performance of the SDP relaxation. A corollary of our results shows that in certain regimes the SDP solutions are in fact integral and exact, improving on existing exact recovery results for convex relaxations. More generally, our results establish sufficient conditions for the SDP to correctly recover the cluster memberships of @math fraction of the points for any @math . As a special case, we show that under the @math -dimensional Stochastic Ball Model, SDP achieves non-trivial (sometimes exact) recovery when the center separation is as small as @math , which complements previous exact recovery results that require constant separation. A theoretical and computational framework for computing intrinsic distance functions and geodesics on submanifolds of @math given by point clouds is introduced and developed in this paper. The basic idea is that, as shown here, intrinsic distance functions and geodesics on general co-dimension submanifolds of @math can be accurately approximated by extrinsic Euclidean ones computed inside a thin offset band surrounding the manifold. This permits the use of computationally optimal algorithms for computing distance functions in Cartesian grids. We use these algorithms, modified to deal with spaces with boundaries, and obtain a computationally optimal approach also for the case of intrinsic distance functions on submanifolds of @math . For point clouds, the offset band is constructed without the need to explicitly find the underlying manifold, thereby computing intrinsic distance functions and geodesics on point clouds while skipping the manifold reconstruction step. The case of poin... We study the question of reconstructing two signals @math and @math from their convolution @math . This problem, known as blind deconvolution , pervades many areas of science and technology, including astronomy, medical imaging, optics, and wireless communications. A key challenge of this intricate non-convex optimization problem is that it might exhibit many local minima. We present an efficient numerical algorithm that is guaranteed to recover the exact solution, when the number of measurements is (up to log-factors) slightly larger than the information-theoretical minimum, and under reasonable conditions on @math and @math . The proposed regularized gradient descent algorithm converges at a geometric rate and is provably robust in the presence of noise. To the best of our knowledge, our algorithm is the first blind deconvolution algorithm that is numerically efficient, robust against noise, and comes with rigorous recovery guarantees under certain subspace conditions. Moreover, numerical experiments do not only provide empirical verification of our theory, but they also demonstrate that our method yields excellent performance even in situations beyond our theoretical framework. The stochastic block model (SBM) with two communities, or equivalently the planted bisection model, is a popular model of random graph exhibiting a cluster behaviour. In the symmetric case, the graph has two equally sized clusters and vertices connect with probability @math within clusters and @math across clusters. In the past two decades, a large body of literature in statistics and computer science has focused on providing lower-bounds on the scaling of @math to ensure exact recovery. In this paper, we identify a sharp threshold phenomenon for exact recovery: if @math and @math are constant (with @math ), recovering the communities with high probability is possible if @math and impossible if @math . In particular, this improves the existing bounds. This also sets a new line of sight for efficient clustering algorithms. While maximum likelihood (ML) achieves the optimal threshold (by definition), it is in the worst-case NP-hard. This paper proposes an efficient algorithm based on a semidefinite programming relaxation of ML, which is proved to succeed in recovering the communities close to the threshold, while numerical experiments suggest it may achieve the threshold. An efficient algorithm which succeeds all the way down to the threshold is also obtained using a partial recovery algorithm combined with a local improvement procedure. The stochastic block model with two communities, or equivalently the planted bisection model, is a popular model of random graph exhibiting a cluster behavior. In the symmetric case, the graph has two equally sized clusters and vertices connect with probability @math within clusters and @math across clusters. In the past two decades, a large body of literature in statistics and computer science has focused on providing lower bounds on the scaling of @math to ensure exact recovery. In this paper, we identify a sharp threshold phenomenon for exact recovery: if @math and @math are constant (with @math ), recovering the communities with high probability is possible if @math and is impossible if $( + 2 ) - . In particular, this improves the existing bounds. This also sets a new line of sight for efficient clustering algorithms. While maximum likelihood (ML) achieves the optimal threshold (by definition), it is in the worst case NP-hard. This paper proposes an efficient algorithm based on a semidefinite programming relaxation of ML, which is proved to succeed in recovering the communities close to the threshold, while numerical experiments suggest that it may achieve the threshold. An efficient algorithm that succeeds all the way down to the threshold is also obtained using a partial recovery algorithm combined with a local improvement procedure. In this paper, we study the mixed linear regression (MLR) problem, where the goal is to recover multiple underlying linear models from their unlabeled linear measurements. We propose a non-convex objective function which we show is locally strongly convex in the neighborhood of the ground truth. We use a tensor method for initialization so that the initial models are in the local strong convexity region. We then employ general convex optimization algorithms to minimize the objective function. To the best of our knowledge, our approach provides first exact recovery guarantees for the MLR problem with @math components. Moreover, our method has near-optimal computational complexity @math as well as near-optimal sample complexity @math for constant @math . Furthermore, we show that our non-convex formulation can be extended to solving the subspace clustering problem as well. In particular, when initialized within a small constant distance to the true subspaces, our method converges to the global optima (and recovers true subspaces) in time linear in the number of points. Furthermore, our empirical results indicate that even with random initialization, our approach converges to the global optima in linear time, providing speed-up of up to two orders of magnitude. A good distance metric is crucial for unsupervised learning from high-dimensional data. To learn a metric without any constraint or class label information, most unsupervised metric learning algorithms appeal to projecting observed data onto a low-dimensional manifold, where geometric relationships such as local or global pairwise distances are preserved. However, the projection may not necessarily improve the separability of the data, which is the desirable outcome of clustering. In this paper, we propose a novel unsupervised adaptive metric learning algorithm, called AML, which performs clustering and distance metric learning simultaneously. AML projects the data onto a low-dimensional manifold, where the separability of the data is maximized. We show that the joint clustering and distance metric learning can be formulated as a trace maximization problem, which can be solved via an iterative procedure in the EM framework. Experimental results on a collection of benchmark data sets demonstrated the effectiveness of the proposed algorithm. The @math -Means clustering problem on @math points is NP-Hard for any dimension @math , however, for the 1D case there exists exact polynomial time algorithms. Previous literature reported an @math time dynamic programming algorithm that uses @math space. It turns out that the problem has been considered under a different name more than twenty years ago. We present all the existing work that had been overlooked and compare the various solutions theoretically. Moreover, we show how to reduce the space usage for some of them, as well as generalize them to data structures that can quickly report an optimal @math -Means clustering for any @math . Finally we also generalize all the algorithms to work for the absolute distance and to work for any Bregman Divergence. We complement our theoretical contributions by experiments that compare the practical performance of the various algorithms. We consider the problem of subspace clustering: given points that lie on or near the union of many low-dimensional linear subspaces, recover the subspaces. To this end, one first identifies sets of points close to the same subspace and uses the sets to estimate the subspaces. As the geometric structure of the clusters (linear subspaces) forbids proper performance of general distance based approaches such as K-means, many model-specific methods have been proposed. In this paper, we provide new simple and efficient algorithms for this problem. Our statistical analysis shows that the algorithms are guaranteed exact (perfect) clustering performance under certain conditions on the number of points and the affinity between subspaces. These conditions are weaker than those considered in the standard statistical literature. Experimental results on synthetic data generated from the standard unions of subspaces model demonstrate our theory. We also show that our algorithm performs competitively against state-of-the-art algorithms on real-world applications such as motion segmentation and face clustering, with much simpler implementation and lower computational cost. In the context of clustering, we assume a generative model where each cluster is the result of sampling points in the neighborhood of an embedded smooth surface; the sample may be contaminated with outliers, which are modeled as points sampled in space away from the clusters. We consider a prototype for a higher-order spectral clustering method based on the residual from a local linear approximation. We obtain theoretical guarantees for this algorithm and show that, in terms of both separation and robustness to outliers, it outperforms the standard spectral clustering algorithm (based on pairwise distances) of Ng, Jordan and Weiss (NIPS '01). The optimal choice for some of the tuning parameters depends on the dimension and thickness of the clusters. We provide estimators that come close enough for our theoretical purposes. We also discuss the cases of clusters of mixed dimensions and of clusters that are generated from smoother surfaces. In our experiments, this algorithm is shown to outperform pairwise spectral clustering on both simulated and real data. Motion planning problems encountered in manipulation and legged locomotion have a distinctive multi-modal structure, where the space of feasible configurations consists of intersecting submanifolds, often of different dimensionalities. Such a feasible space does not possess expansiveness, a property that characterizes whether planning queries can be solved efficiently with traditional probabilistic roadmap (PRM) planners. In this paper we present a new PRM-based multi-modal planning algorithm for problems where the number of intersecting manifolds is finite. We also analyze the completeness properties of this algorithm. More specifically, we show that the algorithm converges quickly when each submanifold is individually expansive and establish a bound on the expected running time in that case. We also present an incremental variant of the algorithm that has the same convergence properties, but works better for problems with a large number of submanifolds by considering subsets of submanifolds likely to contain a solution path. These algorithms are demonstrated in geometric examples and in a legged locomotion planner. Sparse coding has received an increasing amount of interest in recent years. It is an unsupervised learning algorithm, which finds a basis set capturing high-level semantics in the data and learns sparse coordinates in terms of the basis set. Originally applied to modeling the human visual cortex, sparse coding has been shown useful for many applications. However, most of the existing approaches to sparse coding fail to consider the geometrical structure of the data space. In many real applications, the data is more likely to reside on a low-dimensional submanifold embedded in the high-dimensional ambient space. It has been shown that the geometrical information of the data is important for discrimination. In this paper, we propose a graph based algorithm, called graph regularized sparse coding, to learn the sparse representations that explicitly take into account the local manifold structure of the data. By using graph Laplacian as a smooth operator, the obtained sparse representations vary smoothly along the geodesics of the data manifold. The extensive experimental results on image classification and clustering have demonstrated the effectiveness of our proposed algorithm. We consider the problem of clustering a graph @math into two communities by observing a subset of the vertex correlations. Specifically, we consider the inverse problem with observed variables @math , where @math is the incidence matrix of a graph @math , @math is the vector of unknown vertex variables (with a uniform prior), and @math is a noise vector with Bernoulli @math i.i.d. entries. All variables and operations are Boolean. This model is motivated by coding, synchronization, and community detection problems. In particular, it corresponds to a stochastic block model or a correlation clustering problem with two communities and censored edges. Without noise, exact recovery (up to global flip) of @math is possible if and only the graph @math is connected, with a sharp threshold at the edge probability @math for Erdős-Renyi random graphs. The first goal of this paper is to determine how the edge probability @math needs to scale to allow exact recovery in the presence of noise. Defining the degree rate of the graph by @math , it is shown that exact recovery is possible if and only if @math . In other words, @math is the information theoretic threshold for exact recovery at low-SNR. In addition, an efficient recovery algorithm based on semidefinite programming is proposed and shown to succeed in the threshold regime up to twice the optimal rate. For a deterministic graph @math , defining the degree rate as @math , where @math is the minimum degree of the graph, it is shown that the proposed method achieves the rate @math , where @math is the spectral gap of the graph @math . We consider the problem of identifying underlying community-like structures in graphs. Toward this end, we study the stochastic block model (SBM) on k-clusters: a random model on n = km vertices, partitioned in k equal sized clusters, with edges sampled independently across clusters with probability q and within clusters with probability p, p > q. The goal is to recover the initial “hidden” partition of [n]. We study semidefinite programming (SDP)-based algorithms in this context. In the regime (p = (m) m ) and (q = (m) m ), we show that a certain natural SDP-based algorithm solves the problem of exact recovery in the k-community SBM, with high probability, whenever ( - > 1 ), as long as (k=o( n) ). This threshold is known to be the information theoretically optimal. We also study the case when (k= ( (n)) ). In this case however, we achieve recovery guarantees that no longer match the optimal condition ( - > 1 ), thus leaving achieving optimality for this range an open question. A field known as Compressive Sensing (CS) has recently emerged to help address the growing challenges of capturing and processing high-dimensional signals and data sets. CS exploits the surprising fact that the information contained in a sparse signal can be preserved in a small number of compressive (or random) linear measurements of that signal. Strong theoretical guarantees have been established on the accuracy to which sparse or near-sparse signals can be recovered from noisy compressive measurements. In this paper, we address similar questions in the context of a different modeling framework. Instead of sparse models, we focus on the broad class of manifold models, which can arise in both parametric and non-parametric signal families. Building upon recent results concerning the stable embeddings of manifolds within the measurement space, we establish both deterministic and probabilistic instance-optimal bounds in @math for manifold-based signal recovery and parameter estimation from noisy compressive measurements. In line with analogous results for sparsity-based CS, we conclude that much stronger bounds are possible in the probabilistic setting. Our work supports the growing empirical evidence that manifold-based models can be used with high accuracy in compressive signal processing.