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148
Diffusion maps and coarsegraining: A unified framework for dimensionality reduction, graph partitioning and data set parameterization
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2006
"... We provide evidence that nonlinear dimensionality reduction, clustering and data set parameterization can be solved within one and the same framework. The main idea is to define a system of coordinates with an explicit metric that reflects the connectivity of a given data set and that is robust to ..."
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Cited by 158 (5 self)
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We provide evidence that nonlinear dimensionality reduction, clustering and data set parameterization can be solved within one and the same framework. The main idea is to define a system of coordinates with an explicit metric that reflects the connectivity of a given data set and that is robust to noise. Our construction, which is based on a Markov random walk on the data, offers a general scheme of simultaneously reorganizing and subsampling graphs and arbitrarily shaped data sets in high dimensions using intrinsic geometry. We show that clustering in embedding spaces is equivalent to compressing operators. The objective of data partitioning and clustering is to coarsegrain the random walk on the data while at the same time preserving a diffusion operator for the intrinsic geometry or connectivity of the data set up to some accuracy. We show that the quantization distortion in diffusion space bounds the error of compression of the operator, thus giving a rigorous justification for kmeans clustering in diffusion space and a precise measure of the performance of general clustering algorithms.
Random projections of smooth manifolds
 Foundations of Computational Mathematics
, 2006
"... We propose a new approach for nonadaptive dimensionality reduction of manifoldmodeled data, demonstrating that a small number of random linear projections can preserve key information about a manifoldmodeled signal. We center our analysis on the effect of a random linear projection operator Φ: R N ..."
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Cited by 144 (26 self)
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We propose a new approach for nonadaptive dimensionality reduction of manifoldmodeled data, demonstrating that a small number of random linear projections can preserve key information about a manifoldmodeled signal. We center our analysis on the effect of a random linear projection operator Φ: R N → R M, M < N, on a smooth wellconditioned Kdimensional submanifold M ⊂ R N. As our main theoretical contribution, we establish a sufficient number M of random projections to guarantee that, with high probability, all pairwise Euclidean and geodesic distances between points on M are wellpreserved under the mapping Φ. Our results bear strong resemblance to the emerging theory of Compressed Sensing (CS), in which sparse signals can be recovered from small numbers of random linear measurements. As in CS, the random measurements we propose can be used to recover the original data in R N. Moreover, like the fundamental bound in CS, our requisite M is linear in the “information level” K and logarithmic in the ambient dimension N; we also identify a logarithmic dependence on the volume and conditioning of the manifold. In addition to recovering faithful approximations to manifoldmodeled signals, however, the random projections we propose can also be used to discern key properties about the manifold. We discuss connections and contrasts with existing techniques in manifold learning, a setting where dimensionality reducing mappings are typically nonlinear and constructed adaptively from a set of sampled training data.
Protovalue functions: A laplacian framework for learning representation and control in markov decision processes
 Journal of Machine Learning Research
, 2006
"... This paper introduces a novel spectral framework for solving Markov decision processes (MDPs) by jointly learning representations and optimal policies. The major components of the framework described in this paper include: (i) A general scheme for constructing representations or basis functions by d ..."
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Cited by 92 (10 self)
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This paper introduces a novel spectral framework for solving Markov decision processes (MDPs) by jointly learning representations and optimal policies. The major components of the framework described in this paper include: (i) A general scheme for constructing representations or basis functions by diagonalizing symmetric diffusion operators (ii) A specific instantiation of this approach where global basis functions called protovalue functions (PVFs) are formed using the eigenvectors of the graph Laplacian on an undirected graph formed from state transitions induced by the MDP (iii) A threephased procedure called representation policy iteration comprising of a sample collection phase, a representation learning phase that constructs basis functions from samples, and a final parameter estimation phase that determines an (approximately) optimal policy within the (linear) subspace spanned by the (current) basis functions. (iv) A specific instantiation of the RPI framework using leastsquares policy iteration (LSPI) as the parameter estimation method (v) Several strategies for scaling the proposed approach to large discrete and continuous state spaces, including the Nyström extension for outofsample interpolation of eigenfunctions, and the use of Kronecker sum factorization to construct compact eigenfunctions in product spaces such as factored MDPs (vi) Finally, a series of illustrative discrete and continuous control tasks, which both illustrate the concepts and provide a benchmark for evaluating the proposed approach. Many challenges remain to be addressed in scaling the proposed framework to large MDPs, and several elaboration of the proposed framework are briefly summarized at the end.
Value function approximation with diffusion wavelets and Laplacian eigenfunctions
 In Advances in Neural Information Processing Systems 18
, 2006
"... We investigate the problem of automatically constructing efficient representations or basis functions for approximating value functions based on analyzing the structure and topology of the state space. In particular, two novel approaches to value function approximation are explored based on automa ..."
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Cited by 35 (12 self)
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We investigate the problem of automatically constructing efficient representations or basis functions for approximating value functions based on analyzing the structure and topology of the state space. In particular, two novel approaches to value function approximation are explored based on automatically constructing basis functions on state spaces that can be represented as graphs or manifolds: one approach uses the eigenfunctions of the Laplacian, in effect performing a global Fourier analysis on the graph; the second approach is based on diffusion wavelets, which generalize classical wavelets to graphs using multiscale dilations induced by powers of a diffusion operator or random walk on the graph. Together, these approaches form the foundation of a new generation of methods for solving large Markov decision processes, in which the underlying representation and policies are simultaneously learned. 1
Wiener’s lemma for infinite matrices
 Trans. Amer. Math. Soc
, 2006
"... Abstract. The classical Wiener lemma and its various generalizations are important and have numerous applications in numerical analysis, wavelet theory, frame theory, and sampling theory. There are many different equivalent formulations for the classical Wiener lemma, with an equivalent formulation ..."
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Cited by 31 (15 self)
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Abstract. The classical Wiener lemma and its various generalizations are important and have numerous applications in numerical analysis, wavelet theory, frame theory, and sampling theory. There are many different equivalent formulations for the classical Wiener lemma, with an equivalent formulation suitable for our generalization involving commutative algebra of infinite matrices W: = {(a(j − j ′)) j,j ′ ∈Zd: � j∈Zd a(j)  < ∞}. In the study of spline approximation, (diffusion) wavelets and affine frames, Gabor frames on nonuniform grid, and nonuniform sampling and reconstruction, the associated algebras of infinite matrices are extremely noncommutative, but we expect those noncommutative algebras to have a similar property to Wiener’s lemma for the commutative algebra W. In this paper, we consider two noncommutative algebras of infinite matrices, the Schur class and the Sjöstrand class, and establish Wiener’s lemmas for those matrix algebras. 1.
Perfect Reconstruction TwoChannel Wavelet FilterBanks for Graph Structured Data
, 2012
"... In this work we propose the construction of twochannel wavelet filterbanks for analyzing functions defined on the vertices of any arbitrary finite weighted undirected graph. These graph based functions are referred to as graphsignals as we build a framework in which many concepts from the classica ..."
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Cited by 29 (4 self)
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In this work we propose the construction of twochannel wavelet filterbanks for analyzing functions defined on the vertices of any arbitrary finite weighted undirected graph. These graph based functions are referred to as graphsignals as we build a framework in which many concepts from the classical signal processing domain, such as Fourier decomposition, signal filtering and downsampling can be extended to graph domain. Especially, we observe a spectral folding phenomenon in bipartite graphs which occurs during downsampling of these graphs and produces aliasing in graph signals. This property of bipartite graphs, allows us to design critically sampled twochannel filterbanks, and we propose quadrature mirror filters (referred to as graphQMF) for bipartite graph which cancel aliasing and lead to perfect reconstruction. For arbitrary graphs we present a bipartite subgraph decomposition which produces an edgedisjoint collection of bipartite subgraphs. GraphQMFs are then constructed on each bipartite subgraph leading to “multidimensional” separable wavelet filterbanks on graphs. Our proposed filterbanks are critically sampled and we state necessary and sufficient conditions for orthogonality, aliasing cancellation and perfect reconstruction. The filterbanks are realized by Chebychev polynomial approximations.
Geometry of probability spaces
 Constr. Approx
"... Partial differential equations and the Laplacian operator on domains in Euclidean spaces have played a central role in understanding natural phenomena. However this avenue has been limited in many areas where calculus is obstructed as in singular spaces, and function spaces of functions on a space X ..."
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Cited by 22 (1 self)
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Partial differential equations and the Laplacian operator on domains in Euclidean spaces have played a central role in understanding natural phenomena. However this avenue has been limited in many areas where calculus is obstructed as in singular spaces, and function spaces of functions on a space X where X itself is a function space. Examples of the last
Representation Policy Iteration
, 2005
"... This paper addresses a fundamental issue central to approximation methods for solving large Markov decision processes (MDPs): how to automatically learn the underlying representation for value function approximation? A novel theoretically rigorous framework is proposed that automatically generates g ..."
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Cited by 21 (8 self)
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This paper addresses a fundamental issue central to approximation methods for solving large Markov decision processes (MDPs): how to automatically learn the underlying representation for value function approximation? A novel theoretically rigorous framework is proposed that automatically generates geometrically customized orthonormal sets of basis functions, which can be used with any approximate MDP solver like leastsquares policy iteration (LSPI). The key innovation is a coordinatefree representation of value functions, using the theory of smooth functions on a Riemannian manifold. Hodge theory yields a constructive method for generating basis functions for approximating value functions based on the eigenfunctions of the selfadjoint (LaplaceBeltrami) operator on manifolds. In effect, this approach performs a global Fourier analysis on the state space graph to approximate value functions, where the basis functions reflect the largescale topology of the underlying state space. A new class of algorithms called Representation Policy Iteration (RPI) are presented that automatically learn both basis functions and approximately optimal policies. Illustrative experiments compare the performance of RPI with that of LSPI using two handcoded basis functions (RBF and polynomial state encodings).