On Degenerate Doubly Nonnegative Projection Problems

The doubly nonnegative (DNN) cone, being the set of all positive semidefinite matrices whose elements are nonnegative, is a popular approximation of the computationally intractable completely positive cone. The major difficulty for implementing a Newton-type method to compute the projection of a given large-scale matrix onto the DNN cone lies in the possible failure of the constraint nondegeneracy, a generalization of the linear independence constraint qualification for nonlinear programming. Such a failure results in the singularity of the Jacobian of the nonsmooth equation representing the Karush–Kuhn–Tucker optimality condition that prevents the semismooth Newton–conjugate gradient method from solving it with a desirable convergence rate. In this paper, we overcome the aforementioned difficulty by solving a sequence of better conditioned nonsmooth equations generated by the augmented Lagrangian method (ALM) instead of solving one aforementioned singular equation. By leveraging the metric subregularity of the normal cone associated with the positive semidefinite cone, we derive sufficient conditions to ensure the dual quadratic growth condition of the underlying problem, which further leads to the asymptotically superlinear convergence of the proposed ALM. Numerical results on difficult randomly generated instances and from the semidefinite programming library are presented to demonstrate the efficiency of the algorithm for computing the DNN projection to a very high accuracy.

Fischer Type Log-Majorization of Singular Values on Partitioned Positive Semidefinite Matrices

In this paper, we establish a Fischer type log-majorization of singular values on partitioned positive semidefinite matrices, which generalizes the classical Fischer's inequality. Meanwhile, some related and new inequalities are also obtained.

Poisson Quantum Information

By taking a Poisson limit for a sequence of rare quantum objects, I derive simple formulas for the Uhlmann fidelity, the quantum Chernoff quantity, the relative entropy, and the Helstrom information. I also present analogous formulas in classical information theory for a Poisson model. An operator called the intensity operator emerges as the central quantity in the formalism to describe Poisson states. It behaves like a density operator but is unnormalized. The formulas in terms of the intensity operators not only resemble the general formulas in terms of the density operators, but also coincide with some existing definitions of divergences between unnormalized positive-semidefinite matrices. Furthermore, I show that the effects of certain channels on Poisson states can be described by simple maps for the intensity operators.

Fall Detection of Elderly People Using the Manifold of Positive Semidefinite Matrices

Falls are one of the most critical health care risks for elderly people, being, in some adverse circumstances, an indirect cause of death. Furthermore, demographic forecasts for the future show a growing elderly population worldwide. In this context, models for automatic fall detection and prediction are of paramount relevance, especially AI applications that use ambient, sensors or computer vision. In this paper, we present an approach for fall detection using computer vision techniques. Video sequences of a person in a closed environment are used as inputs to our algorithm. In our approach, we first apply the V2V-PoseNet model to detect 2D body skeleton in every frame. Specifically, our approach involves four steps: (1) the body skeleton is detected by V2V-PoseNet in each frame; (2) joints of skeleton are first mapped into the Riemannian manifold of positive semidefinite matrices of fixed-rank 2 to build time-parameterized trajectories; (3) a temporal warping is performed on the trajectories, providing a (dis-)similarity measure between them; (4) finally, a pairwise proximity function SVM is used to classify them into fall or non-fall, incorporating the (dis-)similarity measure into the kernel function. We evaluated our approach on two publicly available datasets URFD and Charfi. The results of the proposed approach are competitive with respect to state-of-the-art methods, while only involving 2D body skeletons.

Indecomposable positive maps on positive semidefinite matrices from Mn to Mn+1

In this paper we obtain a theorem for 2-positive linear maps from Mn(C) to Mn+1(C), where n = 2, 3, 4. In addition, we answer in the affirmative a question that asked if there exists every 2-positive linear map from M3(C) to M4(C) is indecomposable using a family of positive linear maps with Choi matrices of 2-positive maps on positive semidefinite matrices. Further it is shown that 2-positive linear map from M4(C) to M5(C) are indecomposable.

Bad projections of the PSD cone

The image of the cone of positive semidefinite matrices under a linear map is a convex cone. Pataki characterized the set of linear maps for which that image is not closed. The Zariski closure of this set is a hypersurface in the Grassmannian. Its components are the coisotropic hypersurfaces of symmetric determinantal varieties. We develop the convex algebraic geometry of such bad projections, with focus on explicit computations.

Two determinantal inequalities for positive semidefinite matrices

Let A,B,C ∈ Cnxn be positive semidefinite matrices. In this paper, the authors prove two determinantal inequalities|A+B+C|+|C|≥|A+C|+|B+C|+(3n −2 n+1+1)|ABC|1/3and|A+B+C|+|A|+|B|+|C|≥|A+B|+|A+C|+|B+C|+3(3n−1−2n+1)|ABC|1/3.These two inequalities improve known ones.

On Polyhedral Approximations of the Positive Semidefinite Cone

It is well known that state-of-the-art linear programming solvers are more efficient than their semidefinite programming counterparts and can scale to much larger problem sizes. This leads us to consider the question, how well can we approximate semidefinite programs with linear programs? In this paper, we prove lower bounds on the size of linear programs that approximate the positive semidefinite cone. Let D be the set of n × n positive semidefinite matrices of trace equal to one. We prove two results on the hardness of approximating D with polytopes. We show that if 0 < ε < 1and A is an arbitrary matrix of trace equal to one, any polytope P such that (1-ε) (D-A) ⊂ P ⊂ D-A must have linear programming extension complexity at least [Formula: see text], where c > 0 is a constant that depends on ε. Second, we show that any polytope P such that D ⊂ P and such that the Gaussian width of P is at most twice the Gaussian width of D must have extension complexity at least [Formula: see text]. Our bounds are both superpolynomial in n and demonstrate that there is no generic way of approximating semidefinite programs with compact linear programs. The main ingredient of our proofs is hypercontractivity of the noise operator on the hypercube.