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Regularly random duality
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In this paper we look at a class of random optimization problems. We discuss ways that can help determine typical behavior of their solutions. When the dimensions of the optimization problems are large such an information often can be obtained without actually solving the original problems. Moreover, we also discover that fairly often one can actually determine many quantities of interest (such as, for example, the typical optimal values of the objective functions) completely analytically. We present a few general ideas and emphasize that the range of applications is enormous.
Forward citations
Cited by 13 Pith papers
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Precise sample covariance spectral norm error -- an RDT view
For Gaussian data in the proportional limit, the spectral-norm error of the sample covariance converges to γ̂√φ1/(√φ1−√α), with γ̂ solving an equation in the covariance spectrum.
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An RDT based approach to large deviations of Wishart and Wigner matrices spectral edges
Develops an RDT-based LDP framework for spectral edges of Wishart and Wigner matrices matching prior Coulomb gas results.
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Optimal spectral initializers impact on phase retrieval phase transitions -- an RDT view
Optimal spectral initializers at the theoretical phase retrieval threshold sit in flat landscape regions, so roughly 15% oversampling is needed for reliable descending algorithms.
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Phase transition of \emph{descending} phase retrieval algorithms
The paper derives RDT-based lower bounds and predicts a phase transition at oversampling ratio α≈1.4 where descending phase retrieval algorithms transition from failing to succeeding, but the key isomorphism with conv...
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Deep ReLU networks -- injectivity capacity upper bounds
For deep ReLU networks with random Gaussian weights, the paper gives upper bounds on the layer expansion needed for injectivity and finds the expansion need saturates by four layers.
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Controlled Loosening-up (CLuP) -- achieving exact MIMO ML in polynomial time
CLuP, an iterative convex optimization algorithm, is claimed to achieve MIMO ML detection performance in polynomial time, but the claim rests on heuristic random duality arguments and an empirical iteration count.
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A CLuP algorithm to practically achieve $\sim 0.76$ SK--model ground state free energy
The authors propose a CLuP-SK barrier-descent algorithm and report it achieves approximately 0.76 of the SK ground state free energy for n around 2000 to 8000, approaching the theoretical Parisi limit of about 0.763.
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Rare dense solutions clusters in asymmetric binary perceptrons -- local entropy via fully lifted RDT
For the asymmetric binary perceptron, the worst-case local entropy breaks down for constraint density alpha in (0.77, 0.78), matching replica predictions and the range where fast algorithms stop working.
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Phase retrieval with rank $d$ measurements -- \emph{descending} algorithms phase transitions
For rank d phase retrieval with Gaussian measurements, descending gradient algorithms are predicted to succeed above a sample complexity ratio near 2.79 for d=2, with lifted bounds lowering this estimate and simulatio...
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Complexity analysis of the Controlled Loosening-up (CLuP) algorithm
Using Random Duality Theory, the paper argues that the CLuP algorithm reaches near-optimal MIMO ML detection in a small, dimension-independent number of quadratic-programming iterations.
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An RDT based confirmation of Lehner's formula for Kronecker-Gaussian matrices
This paper uses Random Duality Theory to give an alternative proof of Lehner's deterministic spectral edge formula for Kronecker-Gaussian matrices.
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Starting CLuP with polytope relaxation
CLuP-plt, a CLuP detector variant that starts from a box-constrained least-squares solution, reaches near-ML error rates within three to five iterations in the tested MIMO settings.
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High-Dimensional Statistics: Reflections on Progress and Open Problems
This review synthesizes representative advances in high-dimensional statistics, highlights common themes and open problems, and points to key entry works.
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