Spectral dimensionality reduction algorithms are widely used in numerous domains, including for recognition, segmentation, tracking and visualization. However, despite their popularity, these algorithms suffer from a major limitation known as the “repeated eigen-directions” phenomenon. That is, many of the embedding coordinates they produce typically capture the same direction along the data manifold. This leads to redundant and inefficient representations that do not reveal the true intrinsic dimensionality of the data. In this paper, we propose a general method for avoiding redundancy in spectral algorithms. Our approach relies on replacing the orthogonality constraints underlying those methods by unpredictability constraints. Specifically, we require that each embedding coordinate be unpredictable (in the statistical sense) from all previous ones. We prove that these constraints necessarily prevent redundancy, and provide a simple technique to incorporate them into existing methods. As we illustrate on challenging high-dimensional scenarios, our approach produces significantly more informative and compact representations, which improve visualization and classification tasks.
). Here, Projection @math is still redundant. The right column shows the points after subtracting their prediction from previous projections, which causes them to fall off the manifold. (c) The projections obtained with the algorithm of @cite_19 . Here, the algorithm halts after one projection. The right column shows the points after the advection process along the manifold, which results in two clusters forming an unconnected graph. (d) The projections obtained with our non-redundant version of LEM. Our algorithm extracts a non-redundant third projection, which captures progression along the inner angle of the ring.
@cite_19
2129732816 2092349993 2126998703 1950520880
We study the convergence properties of an alternating proximal minimization algorithm for nonconvex structured functions of the type: L(x,y)=f(x)+Q(x,y)+g(y), where f and g are proper lower semicontinuous functions, defined on Euclidean spaces, and Q is a smooth function that couples the variables x and y. The algorithm can be viewed as a proximal regularization of the usual Gauss-Seidel method to minimize L. We work in a nonconvex setting, just assuming that the function L satisfies the Kurdyka-Łojasiewicz inequality. An entire section illustrates the relevancy of such an assumption by giving examples ranging from semialgebraic geometry to “metrically regular” problems. Our main result can be stated as follows: If L has the Kurdyka-Łojasiewicz property, then each bounded sequence generated by the algorithm converges to a critical point of L. This result is completed by the study of the convergence rate of the algorithm, which depends on the geometrical properties of the function L around its critical points. When specialized to @math and to f, g indicator functions, the algorithm is an alternating projection mehod (a variant of von Neumann's) that converges for a wide class of sets including semialgebraic and tame sets, transverse smooth manifolds or sets with “regular” intersection. To illustrate our results with concrete problems, we provide a convergent proximal reweighted l1 algorithm for compressive sensing and an application to rank reduction problems. Chapter 36 Decomposing Graphs into Regions of Small Diameter* Nathan Linialt Michael .lss A decomposition of a graph G = (V, E) is a partition of the vertex set into subsets (called lhks). The diameter of a decomposition is the least. d such that any two vertices belonging to the same connected component of a block are at distance < d. In this paper we prove (nearly best possible) statements of the form: .4ny n–vertex graph has a decomposition into a small number of blocks each having small diameter. Such decompositions provide a tool for efficiently decentralizing distributed computations. In [AGLP1 it was shown that every graph has a decomposition into at most s(n) blocks of diameter at most s(n) for s(n) = o( loglog d h n). usinga,technique of Awerbuch [A] and Awerbuch and Peleg [AP], we improve this result by showing that every graph has a decomposition of diameter ()(log n) into O(log n) blocks. In addition, we give a randomized distributed algorithm that produces such a decomposition and runs in time 0(log2 n). The construction can be parametrized to provide decompositions that trade-off between the number of blocks and the diameter. We show that this trade-off is nearly best possible for two families of graphs the first consists of skeletons of certain triangulations of a simplex and the second consists of grid graphs with added diagonals. The proofs in both cases rely on basic results in combinatorial topology, Sperner’s lemma for the first class and Tucker’s lemma for the second. *This work was supported in part by NSF contracts DMS87-03541 and CCR-8911388 tDepartment of Computer Science, Hebrew University, Jerusalem, Israel. + ‘Department of Computer Science and Engineering, Mail Code C-014, University of California, San Diego, La Jolla, CA 92093-0114. Here, we propose a planning method for knotting unknotting of deformable linear objects. First, we propose a topological description of the state of a linear object. Second, transitions between these states are defined by introducing four basic operations. Then, possible sequences of crossing state transitions, i.e. possible manipulation processes, can be generated once the initial and the objective states are given. Third, a method for determining grasping points and their directions of movement is proposed to realize derived manipulation processes. Our proposed method indicated that it is theoretically possible for any knotting manipulation of a linear object placed on a table to be realized by a one-handed robot with three translational DOF and one rotational DOF. Furthermore, criteria for evaluation of generated plans are introduced to reduce the candidates of manipulation plans. Fourth, a planning method for tying knots tightly is established because they fulfill their fixing function by tightening them. Finally, we report knotting unknotting manipulation performed by a vision-guided system to demonstrate the usefulness of our approach. We cast the problem of motion segmentation of feature trajectories as linear manifold finding problems and propose a general framework for motion segmentation under affine projections which utilizes two properties of trajectory data: geometric constraint and locality. The geometric constraint states that the trajectories of the same motion lie in a low dimensional linear manifold and different motions result in different linear manifolds; locality, by which we mean in a transformed space a data and its neighbors tend to lie in the same linear manifold, provides a cue for efficient estimation of these manifolds. Our algorithm estimates a number of linear manifolds, whose dimensions are unknown beforehand, and segment the trajectories accordingly. It first transforms and normalizes the trajectories; secondly, for each trajectory it estimates a local linear manifold through local sampling; then it derives the affinity matrix based on principal subspace angles between these estimated linear manifolds; at last, spectral clustering is applied to the matrix and gives the segmentation result. Our algorithm is general without restriction on the number of linear manifolds and without prior knowledge of the dimensions of the linear manifolds. We demonstrate in our experiments that it can segment a wide range of motions including independent, articulated, rigid, non-rigid, degenerate, non-degenerate or any combination of them. In some highly challenging cases where other state-of-the-art motion segmentation algorithms may fail, our algorithm gives expected results.