The Experts below are selected from a list of 28722 Experts worldwide ranked by ideXlab platform
Subhabrata Paul - One of the best experts on this subject based on the ideXlab platform.
-
uniformity of point samples in metric spaces using gap ratio
SIAM Journal on Discrete Mathematics, 2017Co-Authors: Arijit Bishnu, Mayank Goswami, Sameer Desai, Arijit Ghosh, Subhabrata PaulAbstract:Teramoto et al. [IEICE Trans. Inform. Syst., 89-D (2006), pp. 2348--2356] defined a measure called the gap ratio that measures the uniformity of a finite point set sampled from $\cal S$, a Bounded Subset of $\mathbb{R}^2$. This definition of uniformity measure can be generalized over all metric spaces by appealing to covering and packing radius. We consider discrete spaces like graph and set of points in the Euclidean space and continuous spaces like the unit square and path connected spaces. The definition of the gap ratio needs only a metric unlike discrepancy, a widely used uniformity measure, that depends on the notion of a range space and its volume. We show some interesting connections of the gap ratio to Delaunay triangulation and packing and covering. Asano [Inform. Process. Lett., 109 (2008), pp. 57--60] opined that the discrete version of the uniformity problem makes it amenable to pose combinatorial optimization related questions. The major focus of this work is on finding lower bounds and solv...
-
uniformity of point samples in metric spaces using gap ratio
Theory and Applications of Models of Computation, 2015Co-Authors: Arijit Bishnu, Mayank Goswami, Sameer Desai, Arijit Ghosh, Subhabrata PaulAbstract:Teramoto et al. [22] defined a new measure called the gap ratio that measures the uniformity of a finite point set sampled from \(\mathcal S\), a Bounded Subset of \(\mathbb {R}^2\). We attempt to generalize the definition of this measure over all metric spaces. We solve optimization related questions about selecting uniform point samples from metric spaces; the uniformity is measured using gap ratio. We give lower bounds for specific metric spaces, prove hardness and approximation hardness results. We also give a general approximation algorithm framework giving different approximation ratios for different metric spaces and give a \(\left( 1+\epsilon \right) \)-approximation algorithm for a set of points in a Euclidean space.
-
uniformity of point samples in metric spaces using gap ratio
arXiv: Computational Geometry, 2014Co-Authors: Arijit Bishnu, Mayank Goswami, Sameer Desai, Arijit Ghosh, Subhabrata PaulAbstract:Teramoto et al. defined a new measure called the gap ratio that measures the uniformity of a finite point set sampled from $\cal S$, a Bounded Subset of $\mathbb{R}^2$. We generalize this definition of measure over all metric spaces by appealing to covering and packing radius. The definition of gap ratio needs only a metric unlike discrepancy, a widely used uniformity measure, that depends on the notion of a range space and its volume. We also show some interesting connections of gap ratio to Delaunay triangulation and discrepancy in the Euclidean plane. The major focus of this work is on solving optimization related questions about selecting uniform point samples from metric spaces; the uniformity being measured using gap ratio. We consider discrete spaces like graph and set of points in the Euclidean space and continuous spaces like the unit square and path connected spaces. We deduce lower bounds, prove hardness and approximation hardness results. We show that a general approximation algorithm framework gives different approximation ratios for different metric spaces based on the lower bound we deduce. Apart from the above, we show existence of coresets for sampling uniform points from the Euclidean space -- for both the static and the streaming case. This leads to a $\left( 1+\epsilon \right)$-approximation algorithm for uniform sampling from the Euclidean space.
A H Tewfik - One of the best experts on this subject based on the ideXlab platform.
-
Bounded Subset selection with noninteger coefficients
European Signal Processing Conference, 2004Co-Authors: Masoud Alghoniemy, A H TewfikAbstract:The Subset selection problem is known to be NP hard. It was recently shown that by relaxing the requirement that the reconstructed signal be equal to the original, one ends with a Bounded error Subset selection that admits a solution in polynomial time. In the Bounded error Subset selection problem, the reconstructed signal is allowed to differ from the original signal by a Bounded error. This Bounded error formulation is natural in many applications, such as coding. In this paper, we improve the accuracy and reduce the complexity of the previously proposed approach for solving the Bounded error Subset selection problem. In particular, unlike the previously proposed approach for solving the Bounded error Subset selection problem, our new algorithm accommodates cases where the coefficients of the closest sparse approximation to the underlying signal in the dictionary are not necessarily one. Our new algorithm is based on weighting the dictionary vectors by the minimum l 2 norm solution and relaxing the integer constraint on the coefficients of the dictionary vectors. It is shown to guarantee high signal accuracy and sparsity. Compared with the Basis Pursuit and the Method of Frames (MoF) algorithms, the proposed algorithm has a better rate-distortion behavior.
-
a sparse solution to the Bounded Subset selection problem a network flow model approach
International Conference on Acoustics Speech and Signal Processing, 2004Co-Authors: Masoud Alghoniemy, A H TewfikAbstract:We reformulate the problem of finding the sparsest representation of a given signal using an overcomplete dictionary as a Bounded error Subset selection problem. Specifically, the reconstructed signal is allowed to differ from the original signal by a Bounded error. We argue that this Bounded error formulation is natural in many applications, such as coding. Our novel formulation guarantees the sparsest solution to the Bounded error Subset selection problem by minimizing the number of nonzero coefficients in the solution vector. We show that this solution can be computed by finding the minimum cost flow path of an equivalent network. Integer programming is adopted to find the solution.
Arijit Bishnu - One of the best experts on this subject based on the ideXlab platform.
-
uniformity of point samples in metric spaces using gap ratio
SIAM Journal on Discrete Mathematics, 2017Co-Authors: Arijit Bishnu, Mayank Goswami, Sameer Desai, Arijit Ghosh, Subhabrata PaulAbstract:Teramoto et al. [IEICE Trans. Inform. Syst., 89-D (2006), pp. 2348--2356] defined a measure called the gap ratio that measures the uniformity of a finite point set sampled from $\cal S$, a Bounded Subset of $\mathbb{R}^2$. This definition of uniformity measure can be generalized over all metric spaces by appealing to covering and packing radius. We consider discrete spaces like graph and set of points in the Euclidean space and continuous spaces like the unit square and path connected spaces. The definition of the gap ratio needs only a metric unlike discrepancy, a widely used uniformity measure, that depends on the notion of a range space and its volume. We show some interesting connections of the gap ratio to Delaunay triangulation and packing and covering. Asano [Inform. Process. Lett., 109 (2008), pp. 57--60] opined that the discrete version of the uniformity problem makes it amenable to pose combinatorial optimization related questions. The major focus of this work is on finding lower bounds and solv...
-
uniformity of point samples in metric spaces using gap ratio
Theory and Applications of Models of Computation, 2015Co-Authors: Arijit Bishnu, Mayank Goswami, Sameer Desai, Arijit Ghosh, Subhabrata PaulAbstract:Teramoto et al. [22] defined a new measure called the gap ratio that measures the uniformity of a finite point set sampled from \(\mathcal S\), a Bounded Subset of \(\mathbb {R}^2\). We attempt to generalize the definition of this measure over all metric spaces. We solve optimization related questions about selecting uniform point samples from metric spaces; the uniformity is measured using gap ratio. We give lower bounds for specific metric spaces, prove hardness and approximation hardness results. We also give a general approximation algorithm framework giving different approximation ratios for different metric spaces and give a \(\left( 1+\epsilon \right) \)-approximation algorithm for a set of points in a Euclidean space.
-
uniformity of point samples in metric spaces using gap ratio
arXiv: Computational Geometry, 2014Co-Authors: Arijit Bishnu, Mayank Goswami, Sameer Desai, Arijit Ghosh, Subhabrata PaulAbstract:Teramoto et al. defined a new measure called the gap ratio that measures the uniformity of a finite point set sampled from $\cal S$, a Bounded Subset of $\mathbb{R}^2$. We generalize this definition of measure over all metric spaces by appealing to covering and packing radius. The definition of gap ratio needs only a metric unlike discrepancy, a widely used uniformity measure, that depends on the notion of a range space and its volume. We also show some interesting connections of gap ratio to Delaunay triangulation and discrepancy in the Euclidean plane. The major focus of this work is on solving optimization related questions about selecting uniform point samples from metric spaces; the uniformity being measured using gap ratio. We consider discrete spaces like graph and set of points in the Euclidean space and continuous spaces like the unit square and path connected spaces. We deduce lower bounds, prove hardness and approximation hardness results. We show that a general approximation algorithm framework gives different approximation ratios for different metric spaces based on the lower bound we deduce. Apart from the above, we show existence of coresets for sampling uniform points from the Euclidean space -- for both the static and the streaming case. This leads to a $\left( 1+\epsilon \right)$-approximation algorithm for uniform sampling from the Euclidean space.
Dominguez T Benavides - One of the best experts on this subject based on the ideXlab platform.
-
a fixed point characterization of weak compactness in banach spaces with unconditional schauder basis
Journal of Mathematical Analysis and Applications, 2017Co-Authors: Dominguez T Benavides, Miguel A JaponAbstract:Abstract Let X be a Banach space with an 1-unconditional Schauder basis and without isomorphic copies of l 1 ⊕ c 0 . We obtain an equivalent condition to weak compactness by means of a fixed-point theorem. Namely: a closed convex Bounded Subset C of X is weakly compact if and only if every cascading nonexpansive mapping T : C → C has a fixed point. We particularize our results when C is the closed unit ball of the Banach space X, obtaining a new characterization of reflexivity. Note that weak compactness is independent of the underlying equivalent norm and that every Banach space with an unconditional Schauder basis can be renormed to be 1-unconditional.
-
weak compactness and fixed point property for affine mappings
Journal of Functional Analysis, 2004Co-Authors: Dominguez T Benavides, M Japon A Pineda, Stanislaw PrusAbstract:Abstract It is shown that a closed convex Bounded Subset of a Banach space is weakly compact if and only if it has the generic fixed point property for continuous affine mappings. The class of continuous affine mappings can be replaced by the class of affine mappings which are uniformly Lipschitzian with some constant M>1 in the case of c0, the class of affine mappings which are uniformly Lipschitzian with some constant M> 6 in the case of quasi-reflexive James’ space J and the class of nonexpansive affine mappings in the case of L-embedded spaces.
Gabor Lugosi - One of the best experts on this subject based on the ideXlab platform.
-
the minimax distortion redundancy in empirical quantizer design
1997Co-Authors: Peter L Bartlett, Tamas Linder, Gabor LugosiAbstract:We obtain minimax lower and upper bounds for the expected distortion redundancy of empirically designed vector quantizers. We show that the mean squared distortion of a vector quantizer designed from data points using any design algorithm is at least away from the optimal distortion for some distribution on a Bounded Subset of . Together with existing upper bounds this result shows that the minimax distortion redundancy for empirical quantizer design, as a function of the size of the training data, is asymptotically on the order of . We also derive a new upper bound for the performance of the empirically optimal quantizer.
-
the minimax distortion redundancy in empirical quantizer design
1997Co-Authors: Peter L Bartlett, Tamas Linder, Gabor LugosiAbstract:We obtain minimax lower and upper bounds for the expected distortion redundancy of empirically designed vector quantizers. We show that the mean squared distortion of a vector quantizer designed from $n$ i.i.d. data points using any design algorithm is at least $\Omega (n^{-1/2})$ away from the optimal distortion for some distribution on a Bounded Subset of ${\cal R}^d$. Together with existing upper bounds this result shows that the minimax distortion redundancy for empirical quantizer design, as a function of the size of the training data, is asymptotically on the order of $n^{1/2}$. We also derive a new upper bound for the performance of the empirically optimal quantizer.