Minimal submanifolds in matrix spaces proven for specific ranks.
problem Minimal submanifolds in matrix spaces.
method Proving semialgebraic sets of matrices are minimal.
result Rectangular, skew-symmetric, and symmetric matrices with prescribed eigenvalues are minimal.
Study inequalities for singular values of rectangular matrices.
problem Inequalities for singular values of rectangular matrices.
method Study convex cones associated to isotropic representations of symmetric spaces.
result Describe inequalities by cohomological conditions.
New framework for higher-order singular-value derivatives of rectangular matrices.
problem Challenging to derive higher-order Fréchet derivatives of singular values in real rectangular matrices.
method Using Kato's analytic perturbation theory for self-adjoint operators and embedding rectangular matrices into block self-adjoint operators.
result Closed-form expressions for the n-th order spectral variations of singular values. Improves signal detection in non-Gaussian noise using transformed data.
problem Signal detection in rank-one signal-plus-noise data matrices.
method Pre-transforming matrix entries and using linear spectral statistics for hypothesis testing.
result Sharp phase transition of largest eigenvalues in spiked rectangular matrices.
Study extends bounds on sample covariance matrices with general dependence.
problem Quantitative bounds on sample covariance matrices with i.i.d. columns.
method Extends previous work on deterministic equivalent to rectangular random matrices with general dependence structure.
result Proves quantitative bounds involving dimensions and spectral parameter, including closer proximity to real positive semi-line.
Computes isotropy subgroups of orthogonal matrices acting on Hermitian matrices.
problem Computing isotropy subgroups of orthogonal matrices acting on Hermitian matrices.
method Algorithm for solving a matrix equation to compute isotropy subgroups.
result Computed isotropy subgroups of orthogonal matrices acting on Hermitian matrices.
Study isotropy groups for complex orthogonal and skew-symmetric matrices.
problem Understanding isotropy subgroups of orthogonal similarity transformations.
method Analysis of group structure of nonsingular block matrices.
result Group structure of isotropy subgroups related to block Toeplitz matrices.
Continuing the quest for exclusive Racah matrices, which are needed for evaluation of colored arborescent-knot polynomials in Chern-Simons theory, we suggest to extract them from a new kind of a double-evolution -- that of the antiparallel double-braids, which is a simple two-parametric family of two-bridge knots, gene…
Algorithm finds isotropy subgroups of orthogonal similarity on symmetric matrices.
problem Computing isotropy subgroups of orthogonal similarity on symmetric matrices.
method Algorithmic procedure solving a Toeplitz matrix equation.
result Structure of isotropy subgroups described.
We elaborate on the recent observation that evolution for twist knots simplifies when described in terms of triangular evolution matrix B, not just its eigenvalues Λ, and provide a universal formula for B, applicable to arbitrary rectangular representation R=[rs]. This expression is in terms of s…
The (stochastic) gradient descent and the multiplicative update method are probably the most popular algorithms in machine learning. We introduce and study a new regularization which provides a unification of the additive and multiplicative updates. This regularization is derived from an hyperbolic analogue of the entr…
Non-symmetric rectangular correlation matrices occur in many problems in economics. We test the method of extracting statistically meaningful correlations between input and output variables of large dimensionality and build a toy model for artificially included correlations in large random time series.The results are t…
Next step is reported in the program of Racah matrices extraction from the differential expansion of HOMFLY polynomials for twist knots: from the double-column rectangular representations R=[rr] to a triple-column and triple-hook R=[333]. The main new phenomenon is the deviation of the particular coefficient $f_{[332]}…
KNTZ trick simplifies knot polynomial calculations for twist knots.
problem Completing the structure of differential expansion for twist knots.
method Converting arborescent evolution matrix into triangular form.
result Conjecture for triangular matrix B in non-rectangular case. Many knots and links in S^3 can be drawn as gluing of three manifolds with one or more four-punctured S^2 boundaries. We call these knot diagrams as double fat graphs whose invariants involve only the knowledge of the fusion and the braiding matrices of four-strand braids. Incorporating the properties of four-point con…
We describe the inclusive Racah matrices for the first non-(anti)symmetric rectangular representation R=[2,2] for quantum groups U_q(sl_N). Most of them have sizes 2, 3, and 4 and are fully described by the eigenvalue hypothesis. Of two 6x6 matrices, one is also described in this way, but the other one corresponds to t…
We present a general method to detect and extract from a finite time sample statistically meaningful correlations between input and output variables of large dimensionality. Our central result is derived from the theory of free random matrices, and gives an explicit expression for the interval where singular values are…
Rectangular diagrams help analyze foliations in 3-sphere.
problem Analyzing foliations in 3-sphere complements.
method Introduced rectangular diagrams for foliations and links.
result Any co-orientable finite depth foliation can be presented by a compatible rectangular diagram.
The paper studies phase transitions in random matrices and tensor unfolding for detecting signals.
problem Phase transitions in singular values and vectors of large random matrices.
method Analysis of singular values and vectors of long rectangular random matrices, and tensor unfolding algorithm for asymmetric rank-one spiked tensor models.
result An exact threshold for tensor unfolding to detect signals, independent of unfolding procedure.
Random feature matrices' singular values concentrate near their full expectation in high dimensions.
problem Characterizing the spectra of random feature matrices for regression problems.
method Analyzing two settings of input variables (random or well-separated) with conditions on dimension, complexity ratio, and sampling variance.
result The singular values of random feature matrices concentrate near their full expectation and near one with high probability.
GD and NAG accelerate matrix factorization and neural networks.
problem Optimizing rectangular matrix factorization and linear neural networks.
method Gradient descent and Nesterov's accelerated gradient with specific initialization.
result NAG achieves the best-known iteration complexity for these problems.
Mirror descent algorithm recovers low-rank matrices in matrix sensing.
problem Matrix sensing with low-rank matrices under certain conditions.
method Discrete-time mirror descent applied to empirical risk with Bregman divergence analysis.
result Mirror descent converges to a matrix minimizing a specific nuclear norm-related quantity.
Improved bounds for knot crossings in different mosaic patterns.
problem Finding tighter bounds for knot crossings in rectangular and hexagonal mosaics.
method Extended Howard and Kobin's proof to hexagonal mosaics and shortened the rectangular proof.
result New bounds for hexagonal mosaics with improved efficiency in rectangular mosaics.
Study heavy-tailed weights' impact on neural network's spectral distribution.
problem Analyzing spectral distribution of conjugate kernel matrices with heavy-tailed weights.
method Computed limiting eigenvalue distribution through moments, considering heavy-tailed distributions and nonlinear activation functions.
result Heavy-tailed weights induce strong correlations, leading to fundamentally different spectral behavior.
The paper proves a distribution claim for neural network Jacobians.
problem Distribution of singular values in deep neural networks.
method Free probability and random matrix theory techniques.
result Singular value distribution matches for specific cases.
New proof for symmetric spaces with rectangular lattices.
problem Characterizing symmetric spaces with rectangular unit lattices.
method Explicit construction of isometric embeddings and analysis of root systems.
result Symmetric spaces with rectangular unit lattices are symmetric R-spaces.
Rectangular peg problem solved for many curves.
problem Rectangular peg problem for continuous Jordan curves.
method Microlocal sheaf theory and recent work of Greene and Lobb.
result Affirmative answer for a large class of rectifiable curves.
Proves a theorem for comparing surfaces in 3D space.
problem Comparing isotopy classes of compact surfaces in 3-sphere.
method Uses rectangular diagrams to formalize and compare surfaces.
result Proves a Reidemeister type theorem for rectangular diagrams of surfaces.
If a rectangular diagram represents the trivial knot, then it can be deformed into the rectangular diagram with only two vertical edges by a finite sequence of merge operations and exchange operations, without increasing the number of vertical edges, which was shown by I. A. Dynnikov. We show in this paper that we need…
In this paper Legendrian graphs in (R3,ξst) are considered modulo Legendrian isotopy and edge contraction. To a Legendrian graph we associate a (generalized) rectangular diagram --- a purely combinatorial object. Moves of rectangular diagrams are introduced so that equivalence classes of Legendr…
Paper studies S-rectangular DR-RL models for robust reinforcement learning with near-optimal sample complexity.
problem Addressing distributional discrepancies in reinforcement learning environments.
method Empirical value iteration algorithm for divergence-based S-rectangular DR-RL models.
result Near-optimal sample complexity bound of O(∣S∣∣A∣(1−γ)−4ε−2). The study improves inequalities for link diagrams and introduces weak rectangular diagrams.
problem Improving inequalities for link diagrams and understanding their properties.
method Introducing weak rectangular diagrams and proving new inequalities.
result Generalizes and subsumes many known inequalities related to multi-crossing numbers.
We introduce a simple combinatorial way, which we call a rectangular diagram of a surface, to represent a surface in the three-sphere. It has a particularly nice relation to the standard contact structure on S3 and to rectangular diagrams of links. By using rectangular diagrams of surfaces we are going, in p…
Rectangular mosaics extend virtual knot studies to larger polygons.
problem Studying virtual knots using mosaic techniques.
method Introduced rectangular mosaics, modified mosaic moves, and provided invariants.
result Developed algorithms for computing virtual knot invariants.
Study reveals 1/f noise in signals made from nonoverlapping rectangular pulses.
problem Analyzing 1/f noise in signals composed of nonoverlapping pulses. method Derived a general formula for power spectral density, analyzed rectangular pulse case.
result Observed pure 1/f noise until very low frequencies with long pulse durations. We claim that the recently discovered universal-matrix precursor for the F functions, which define the differential expansion of colored polynomials for twist and double braid knots, can be extended from rectangular to non-rectangular representations. This case is far more interesting, because it involves multiplicit…
We conjecture a closed-form expression of HOMFLY-PT invariants of double twist knots colored by rectangular Young diagrams where the twist is encoded in interpolation Macdonald polynomials. We also put forth a conjecture of cyclotomic expansions of HOMFLY-PT polynomials colored by rectangular Young diagrams for any kno…
Study on overlaps of singular vectors in Gaussian matrix submatrices.
problem Analyzing overlaps of singular vectors in submatrices of Gaussian matrices.
method Utilizes dynamics of singular vectors and specific resolvents for Brownian trajectories.
result Explicit forms for limiting rescaled mean squared overlaps in the bulk of spectra.
If a rectangular diagram represents the trivial knot, then it can be deformed into the trivial rectangular diagram with only four edges by a finite sequence of merge operations and exchange operations, without increasing the number of edges, which was shown by I. A. Dynnikov. Using this, Henrich and Kauffman gave an up…
A correspondence is studied by H. Matsuda between front projections of Legendrian links in the standard contact structure for 3-space and rectangular diagrams. In this paper, we introduce braided rectangular diagrams, and study a relationship with Legendrian links in the standard contact structure for 3-space. We show …
New transformations preserve link isotopy, showing complexity differences.
problem Link isotopy preservation with complexity differences.
method Introducing multiflypes of rectangular diagrams of links.
result Two diagrams of same complexity not related by simpler moves.
Paper studies robust MDPs, improving sample complexity and asymptotic performance.
problem Optimal robust policy and value function in robust MDPs with generative models.
method Improves prior results on non-asymptotic and asymptotic performances of robust MDPs, considering various uncertainty sets.
result Improved sample complexity and asymptotic normality of optimal robust value function.
Deep learning classifies knots using rectangular diagrams.
problem Recognizing and distinguishing knots, especially the unknot.
method Represent knots as rectangular Dynnikov diagrams and use neural networks to classify them.
result Neural networks can effectively distinguish knots from each other.
Factorization of the differential expansion coefficients for HOMFLY-PT polynomials of double braids, discovered in arXiv:1606.06015 in the case of rectangular representations R, is extended to the first non-rectangular representations R=[2,1] and R=[3,1]. This increases chances that such factorization will take p…
In the present paper a criteria for a rectangular diagram to admit a simplification is given in terms of Legendrian knots. It is shown that there are two types of simplifications which are mutually independent in a sense. A new proof of the monotonic simplification theorem for the unknot is given. It is shown that a mi…
Optimal unimodal fitting for linear loss functions in a sequential, efficient manner.
problem Optimal unimodal transformation of univariate model scores under linear loss functions.
method Proposes a sequential approach to estimate the optimal rectangular fit for observed samples with each new sample.
result Sequential approach achieves optimal efficiency with logarithmic time complexity per iteration.
The paper proves that any smooth curve can have two similar inscribed rectangles.
problem Finding two similar inscribed rectangles in a smooth Jordan curve.
method Lagrangian Floer homology and differential topological computation.
result Generic doubling of inscribed rectangles in smooth Jordan curves.
Square Clifford torus uniquely determined by isoperimetric ratio, rectangular torus not.
problem Uniqueness of 3D shape of rectangular Clifford torus based on isoperimetric ratio.
method Closed-form formulas for isoperimetric ratio of stereographic projection, strict monotonicity.
result Isoperimetric ratio does not uniquely determine rectangular Clifford torus shape.