Ky Fan Norms Research Articles

Video fragment processing by Ky Fan norm

In this study, we focused on the formalization of video frame descriptions in the context of solving video segmentation problem. Since native video data can have various sizes, dividing each frame into blocks allows present image frame as a square matrix for a formal description. The frame block is a matrix of arbitrary dimensions. The ability to skip the step of matrix transformation to a square dimension or vectorization using some descriptor allows to reduce computational costs, freeing up computational resources required for this transformation. In our study, we used Ky Fan norm value as image frame block descriptor. The Ky Fan norm is built on top of matrix singular values. A singular decomposition does not impose restrictions on either the dimension orthe character of the elements of the original matrix. We conducted a comparative analysis of the effectiveness of the obtained descriptor for different video data sizesand with different aspect ratios, showing that the change in the descriptor for each block is independent of the video sizeand aspect ratios. Changes in the descriptorsfor each block from frame to frame are identical for video data of varying sizes. This means that as a result of such fragment transform, a square matrix of a fixed size iscreated, regardless of the output video size.This makes it possible to unify further processing of the video, which can be useful for the task of information search in large video databases under the conditions of providing a query "ad exemplum". In thiscase, we can analyze the existing database in offline mode and match each video with a fixed square matrix of descriptors, which will significantly reduce the time and amount of resources when matching with the query. Also, this approach can be effectively used to analyze video data for the motion detection and scene change tracking.

On the Ky Fan norm of the signless Laplacian matrix of a graph

On the Ky Fan norm of the signless Laplacian matrix of a graph

A family of separability criteria and lower bounds of concurrence

The problem on detecting the entanglement of a bipartite state is significant in quantum information theory. In this article, we apply the Ky Fan norm to the revised realignment matrix of a bipartite state. We consider a family of separable criteria for bipartite states and present when the density matrix corresponds to a state is real, the criterion is equivalent to the enhanced realignment criterion. Moreover, we present analytical lower bounds of concurrence and the convex-roof extended negativity for arbitrary dimensional systems.

Minimax perturbation bounds of the low-rank matrix under Ky Fan norm

<abstract><p>This paper considers the minimax perturbation bounds of the low-rank matrix under Ky Fan norm. We first explore the upper bounds via the best rank-$ r $ approximation $ \hat{A}_r $ of the observation matrix $ \hat{A} $. Next, the lower bounds are established by constructing special matrix groups to show the upper bounds are tight on the low-rank matrix estimation error. In addition, we derive the rate-optimal perturbation bounds for the left and right singular subspaces under Ky Fan norm $ \sin\Theta $ distance. Finally, some simulations have been carried out to support our theories.</p></abstract>

Combo separability criteria and lower bound on concurrence

Detection and quantification of entanglement are extremely important in quantum information theory. We can extract information by using the spectrum or singular values of the density operator. The correlation matrix norm deals with the concept of quantum entanglement in a mathematically natural way. In this work, we use Ky Fan norm of the Bloch matrix to investigate the disentanglement of quantum states. Our separability criterion not only unifies some well-known criteria but also leads to a better lower bound on concurrence. We explain with an example how the entanglement of the given state is missed by existing criteria but can be detected by our criterion. The proposed lower bound on concurrence also has advantages over some investigated bounds.

A novel nonlinear dimension reduction approach to infer population structure for low-coverage sequencing data

BackgroundLow-depth sequencing allows researchers to increase sample size at the expense of lower accuracy. To incorporate uncertainties while maintaining statistical power, we introduce MCPCA_PopGen to analyze population structure of low-depth sequencing data.ResultsThe method optimizes the choice of nonlinear transformations of dosages to maximize the Ky Fan norm of the covariance matrix. The transformation incorporates the uncertainty in calling between heterozygotes and the common homozygotes for loci having a rare allele and is more linear when both variants are common.ConclusionsWe apply MCPCA_PopGen to samples from two indigenous Siberian populations and reveal hidden population structure accurately using only a single chromosome. The MCPCA_PopGen package is available on https://github.com/yiwenstat/MCPCA_PopGen.

KY FAN NORM

KY FAN NORM

Ky Fan norm application for video segmentation

This article presents results of applying the KyFan norm in the context of solving the problem of video segmentation. Since the task of video analysis can be considered as analysis of the sequence of images, it was decided to find a way to formalize the description of the video frame using the mathematical apparatus of non-square matrices. When choosing a method, particular attention was paid precisely to universality with respect to the dimension of the initial data due to the technical characteristics and nature of the video data -video frames are matrices of arbitrary dimension. The ability to skip the step of matrix transformation to square dimension, or vectorization using some descriptor allows you to reduce computational costsrequired for this transformation. It was decided to use the value of the Ky Fan norm as an image descriptor, since it is built on top of matrix singular values. As it is known, singular values are calculated during the singular decomposition of the matrix and can be used, among other features, to reduce the dimension of the source data. A singular decomposition does not impose restrictions on either the dimension or the character of the elements of the original matrix. In addition, it can be used to derive other matrix decompositions with required characteristics. A comparative analysis of the effectiveness of the obtained descriptor was carried out in the case of using the k-norm and 1-norm, which showed that the 1-norm allows us to identify the most significant changes in the scene, while k -norm is able to detect minor. In other words, depending on the context of the source video data and the scope of the developed application, it is possible to configure the sensitivity of the application to a change in the scene by varying thenumber of singular values involved. The decision about the presence of changes in the context of video scene is made based on a comparison of descriptors of two consecutive images, that is, the values of the Ky Fan norm.

Maximally Correlated Principal Component Analysis Based on Deep Parameterization Learning

Dimensionality reduction is widely used to deal with high-dimensional data. As a famous dimensionality reduction method, principal component analysis (PCA) aiming at finding the low dimension feature of original data has made great successes, and many improved PCA algorithms have been proposed. However, most algorithms based on PCA only consider the linear correlation of data features. In this article, we propose a novel dimensionality reduction model called maximally correlated PCA based on deep parameterization learning (MCPCADP), which takes nonlinear correlation into account in the deep parameterization framework for the purpose of dimensionality reduction. The new model explores nonlinear correlation by maximizing Ky-Fan norm of the covariance matrix of nonlinearly mapped data features. A new BP algorithm for model optimization is derived. In order to assess the proposed method, we conduct experiments on both a synthetic database and several real-world databases. The experimental results demonstrate that the proposed algorithm is comparable to several widely used algorithms.

Maps preserving a Ky-Fan norm of Jordan product

Given n≥2 let 2≤k≤n be fixed. We study mappings ϕ on Mn, the algebra of n×n complex matrices, that satisfy Nk(ϕ(A)ϕ(B)+ϕ(B)ϕ(A))=Nk(AB+BA) for all A, B∈Mn, where Nk(C) denotes the Ky-Fan k-norm, the sum of k greatest singular values of the matrix C. If n≥3, we additionally suppose either ϕ(μ1I)=μ2I, for some unimodular complex μ1, μ2, or, that ϕ is surjective; and when n=2, the complete description is obtained without additional assumptions.

Extrema of graph eigenvalues

In 1993 Hong asked what are the best bounds on the k'th largest eigenvalue λk(G) of a graph G of order n. This challenging question has never been tackled for any 2<k<n. In the present paper tight bounds are obtained for all k>2, and even tighter bounds are obtained for the k'th largest singular value λk⁎(G).Some of these bounds are based on Taylor's strongly regular graphs, and others on a method of Kharaghani for constructing Hadamard matrices. The same kind of constructions are applied to other open problems, like Nordhaus–Gaddum problems of the kind: How large canλk(G)+λk(G¯)be?These constructions are successful also in another open question: How large can the Ky Fan normλ1⁎(G)+⋯+λk⁎(G)be? Ky Fan norms of graphs generalize the concept of graph energy, so this question generalizes the problem for maximum energy graphs.In the final section, several results and problems are restated for (−1,1)-matrices, which seem to provide a more natural ground for such research than graphs.Many of the results in the paper are paired with open questions and problems for further study.

Maximum norms of graphs and matrices, and their complements

Given a graph G, let G∗ denote the trace norm of its adjacency matrix, also known as the energy of G. The main result of this paper states that if G is a graph of order n, thenG∗+G¯∗⩽n-11+n,where G¯ is the complement of G. Equality is possible if and only if G is a strongly regular graph with parameters n,n-1/2,n-5/4,n-1/4, known also as a conference graph.In fact, the above problem is stated and solved in a more general setup - for nonnegative matrices with bounded entries. In particular, this study exhibits analytical matrix functions attaining maxima on matrices with rigid and complex combinatorial structure.In the last section the same questions are studied for Ky Fan norms. Possibe directions for further research are outlined, as it turns out that the above problems are just a tip of a larger multidimensional research area.

Linear Maps Preserving Ky Fan Norms and Schatten Norms of Tensor Products of Matrices

For a positive integer $n$, let $M_n$ be the set of $n\times n$ complex matrices. Suppose $\|\cdot\|$ is the Ky Fan $k$-norm with $1 \le k \le mn$ or the Schatten $p$-norm with $1 \le p \le \infty$ ($p\ne 2$) on $M_{mn}$, where $m,n\ge 2$ are positive integers. It is shown that a linear map $\phi: M_{mn} \rightarrow M_{mn}$ satisfying $\|A\otimes B\| = \|\phi(A\otimes B)\| for all A \in M_m and B \in M_n$ if and only if there are unitary $U, V \in M_{mn}$ such that $\phi$ has the form $A\otimes B \mapsto U(\varphi_1(A) \otimes \varphi_2(B))V$, where $\varphi_s(X)$ is either the identity map $X \mapsto X$ or the transposition map $X \mapsto X^t$. The results are extended to tensor space $M_{n_1} \otimes \cdots \otimes M_{n_m}$ of higher level. The connection of the problem to quantum information science is mentioned.

Relations for Certain Symmetric Norms and Anti-norms Before and After Partial Trace

Changes of some unitarily invariant norms and anti-norms under the operation of partial trace are examined. The norms considered form a two-parametric family, including both the Ky Fan and Schatten norms as particular cases. The obtained results concern operators acting on the tensor product of two finite-dimensional Hilbert spaces. For any such operator, we obtain upper bounds on norms of its partial trace in terms of the corresponding dimensionality and norms of this operator. Similar inequalities, but in the opposite direction, are obtained for certain anti-norms of positive matrices. Through the Stinespring representation, the results are put in the context of trace-preserving completely positive maps. We also derive inequalities between the unified entropies of a composite quantum system and one of its subsystems, where traced-out dimensionality is involved as well.

Low rank multivariate regression

We consider in this paper the multivariate regression problem, when the target regression matrix A is close to a low rank matrix. Our primary interest is in on the practical case where the variance of the noise is unknown. Our main contribution is to propose in this setting a criterion to select among a family of low rank estimators and prove a non-asymptotic oracle inequality for the resulting estimator. We also investigate the easier case where the variance of the noise is known and outline that the penalties appearing in our criterions are minimal (in some sense). These penalties involve the expected value of Ky-Fan norms of some random matrices. These quantities can be evaluated easily in practice and upper-bounds can be derived from recent results in random matrix theory.

Bounds on Shannon distinguishability in terms of partitioned measures

A family of quantum measures like the Shannon distinguishability is presented. These measures are defined over the two classes of POVM measurements and related to separate parts in the expression for mutual information. Changes of Ky Fan's norms and the partitioned trace distances under the operation of partial trace are discussed. Upper and lower bounds on the introduced quantities are obtained in terms of partitioned trace distances and Uhlmann's partial fidelities. These inequalities provide a kind of generalization of the well-known bounds on the Shannon distinguishability. The notion of cryptographic exponential indistinguishability for quantum states is revisited. When exponentially fast convergence is required, all the metrics induced by unitarily invariant norms are shown to be equivalent.

Partitioned trace distances

New quantum distance is introduced as a half-sum of several singular values of difference between two density operators. This is, up to factor, the metric induced by so-called Ky Fan norm. The partitioned trace distances enjoy similar properties to the standard trace distance, including the unitary invariance, the strong convexity and the close relations to the classical distances. The partitioned distances cannot increase under quantum operations of certain kind including bistochastic maps. All the basic properties are re-formulated as majorization relations. Possible applications to quantum information processing are briefly discussed.

Tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property

In this paper we set up a representation theorem for tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property in terms of Ky Fan norms. Examples of tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property include unitarily invariant norms on finite factors (type II 1 factors and M n ( C ) ) and symmetric gauge norms on L ∞ [ 0 , 1 ] and C n . As the first application, we obtain that the class of unitarily invariant norms on a type II 1 factor coincides with the class of symmetric gauge norms on L ∞ [ 0 , 1 ] and von Neumann's classical result [J. von Neumann, Some matrix-inequalities and metrization of matrix-space, Tomsk. Univ. Rev. 1 (1937) 286–300] on unitarily invariant norms on M n ( C ) . As the second application, Ky Fan's dominance theorem [Ky Fan, Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Natl. Acad. Sci. USA 37 (1951) 760–766] is obtained for finite von Neumann algebras satisfying the weak Dixmier property. As the third application, some classical results in non-commutative L p -theory (e.g., non-commutative Hölder's inequality, duality and reflexivity of non-commutative L p -spaces) are obtained for general unitarily invariant norms on finite factors. We also investigate the extreme points of N ( M ) , the convex compact set (in the pointwise weak topology) of normalized unitarily invariant norms (the norm of the identity operator is 1) on a finite factor M . We obtain all extreme points of N ( M 2 ( C ) ) and some extreme points of N ( M n ( C ) ) ( n ⩾ 3 ). For a type II 1 factor M , we prove that if t ( 0 ⩽ t ⩽ 1 ) is a rational number then the Ky Fan tth norm is an extreme point of N ( M ) .

Dimension Reduction and Coefficient Estimation in Multivariate Linear Regression

SummaryWe introduce a general formulation for dimension reduction and coefficient estimation in the multivariate linear model. We argue that many of the existing methods that are commonly used in practice can be formulated in this framework and have various restrictions. We continue to propose a new method that is more flexible and more generally applicable. The method proposed can be formulated as a novel penalized least squares estimate. The penalty that we employ is the coefficient matrix's Ky Fan norm. Such a penalty encourages the sparsity among singular values and at the same time gives shrinkage coefficient estimates and thus conducts dimension reduction and coefficient estimation simultaneously in the multivariate linear model. We also propose a generalized cross-validation type of criterion for the selection of the tuning parameter in the penalized least squares. Simulations and an application in financial econometrics demonstrate competitive performance of the new method. An extension to the non-parametric factor model is also discussed.

Isometries for Ky Fan norms between matrix spaces

We characterize linear maps between different rectangular matrix spaces preserving Ky Fan norms.