Duality for Nonconvex Approximation and Optimization

Singer, Ivan

doi:10.1007/0-387-28395-1

Cited by 31 publications

(11 citation statements)

References 127 publications

(245 reference statements)

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“…( 11) becomes smaller, which implies that the rate function can only become larger. Thus, since (r, J) was arbitrary, (15) implies R(J) ≥ M (J).…”

Section: A Protocol For the Quantum Information Bottleneck Methodsmentioning

confidence: 99%

“…( 11) is not of convex type due to the sign of the inequality in the constraint (≥ rather than ≤). Just as in the classical case, it is a so-called "reverse convex problem" (see, e.g., [15]), for which many of the standard results, such as strong duality or convexity of the resulting function in J, are not known to hold or apply only in a weaker form. Nevertheless, we present numerical evidence for convexity of the RHS in Appendix B.…”

Section: A Protocol For the Quantum Information Bottleneck Methodsmentioning

confidence: 99%

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Quantum Rate-Distortion Coding of Relevant Information

Salek

Cadamuro

Kammerlander

et al. 2019

IEEE Trans. Inform. Theory

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Rate-distortion theory provides bounds for compressing data produced by an information source to a specified encoding rate that is strictly less than the source's entropy. This necessarily entails some loss, or distortion, between the original source data and the best approximation after decompression. The so-called Information Bottleneck Method is designed to compress only 'relevant' information. Which information is relevant is determined by the correlation between the data being compressed and a variable of interest, so-called side information. In this paper, an Information Bottleneck Method is introduced for the compression of quantum data. The channel communication picture is used for compression and decompression. The rate of compression is derived using an entanglement assisted protocol with classical communication, and under an unproved conjecture that the rate function is convex in the distortion parameter. The optimum channel achieving this rate for a given input state is characterised. The conceptual difficulties arising due to differences in the mathematical formalism between quantum and classical probability theory are discussed and solutions are presented.

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“…( 11) becomes smaller, which implies that the rate function can only become larger. Thus, since (r, J) was arbitrary, (15) implies R(J) ≥ M (J).…”

Section: A Protocol For the Quantum Information Bottleneck Methodsmentioning

confidence: 99%

Section: A Protocol For the Quantum Information Bottleneck Methodsmentioning

confidence: 99%

Quantum Rate-Distortion Coding of Relevant Information

Salek

Cadamuro

Kammerlander

et al. 2019

IEEE Trans. Inform. Theory

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“…Some of our results are inspired by -and bear some analogy with-those known from the theory of best approximation in normed linear spaces by elements of linear subspaces (see e.g. [21]), reformulated in terms of the "semi-scalar product" (see e.g. [18]).…”

Section: Introductionmentioning

confidence: 92%

Best approximation in max-plus semimodules

Akian

Gaubert

Niţică

et al. 2011

Linear Algebra and its Applications

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We establish new results concerning projectors on max-plus spaces, as well as separating half-spaces, and derive an explicit formula for the distance in Hilbert's projective metric between a point and a half-space over the max-plus semiring, as well as explicit descriptions of the set of minimizers. As a consequence, we obtain a cyclic projection type algorithm to solve systems of max-plus linear inequalities.Comment: 42 pages and 4 figure

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“…The key idea of this framework is to globally approximate J using a sequence of quadratic functions [45]. Taking the subproblem over x as an example, after having found x k , we can construct a quadratic function ψ(x; x k ) to upper bound J (x) such that the following conditions hold:…”

Section: Convergence Analysismentioning

confidence: 99%

“…In non-convex optimization [45], a common assumption is that the non-convex function J (·) is "locally convex" around its "local" minimum. Suppose we assume that the initialization x 0 is "good", in that it is situated sufficiently close a local minimizer x * .…”

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confidence: 99%

Multi-View Intact Space Learning

Tao

2015

IEEE Trans. Pattern Anal. Mach. Intell.

380

101

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It is practical to assume that an individual view is unlikely to be sufficient for effective multi-view learning. Therefore, integration of multi-view information is both valuable and necessary. In this paper, we propose the Multi-view Intact Space Learning (MISL) algorithm, which integrates the encoded complementary information in multiple views to discover a latent intact representation of the data. Even though each view on its own is insufficient, we show theoretically that by combing multiple views we can obtain abundant information for latent intact space learning. Employing the Cauchy loss (a technique used in statistical learning) as the error measurement strengthens robustness to outliers. We propose a new definition of multi-view stability and then derive the generalization error bound based on multi-view stability and Rademacher complexity, and show that the complementarity between multiple views is beneficial for the stability and generalization. MISL is efficiently optimized using a novel Iteratively Reweight Residuals (IRR) technique, whose convergence is theoretically analyzed. Experiments on synthetic data and real-world datasets demonstrate that MISL is an effective and promising algorithm for practical applications.

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Duality for Nonconvex Approximation and Optimization

Cited by 31 publications

References 127 publications

Quantum Rate-Distortion Coding of Relevant Information

Quantum Rate-Distortion Coding of Relevant Information

Best approximation in max-plus semimodules

Multi-View Intact Space Learning

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