Matrix methods for perfect signal recovery underlying range space of operators

The purpose of this work is to investigate perfect reconstruction underlying range space of operators in finite dimensional Hilbert spaces by a new matrix method. To this end, first we obtain more structures of the canonical $K$-dual. % and survey optimal $K$-dual problem under probabilistic erasures. Then, we survey the problem of recovering and robustness of signals when the erasure set satisfies the minimal redundancy condition or the $K$-frame is maximal robust. Furthermore, we show that the error rate is reduced under erasures if the $K$-frame is of uniform excess. Toward the protection of encoding frame (K-dual) against erasures, we introduce a new concept so called $(r,k)$-matrix to recover lost data and solve the perfect recovery problem via matrix equations. Moreover, we discuss the existence of such matrices by using minimal redundancy condition on decoding frames for operators. We exhibit several examples that illustrate the advantage of using the new matrix method with respect to the previous approaches in existence construction. And finally, we provide the numerical results to confirm the main results in the case noise-free and test sensitivity of the method with respect to noise.

A layer potential approach to functional and clinical brain imaging

In this work, we consider the inverse source recovery problem from sEEG, EEG and MEG point-wise data. We regard this as an inverse source recovery problem for L2 vector-fields normally oriented and supported on the grey/white matter interface, which together with the brain, skull and scalp form a non-homogeneous layered conductor. We assume that the quasistatic approximation of Maxwell’s equation holds for the electro-magnetic fields considered. The electric data is measured point-wise inside and outside the conductor while the magnetic data is measured only point-wise outside the conductor. These ill-posed problems are solved via Tikhonov regularization on triangulations of the interfaces and a piecewise linear model for the current on the triangles. Both in the continuous and discrete formulation the electric potential is expressed as a linear combination of double layer potentials while the magnetic flux density in the continuous case is a vector-surface integral whose discrete formulation features single layer potentials. A main feature of our approach is that these contributions can be computed exactly. Due to the consideration of the regularity conditions of the electric potential in the inverse source recovery problem, the Cauchy transmission problem for the electric potential is inadvertently solved as well. In the problem, we propagate only the electric potential while the normal derivatives at the interfaces of discontinuity of the electric conductivities are computed directly from the resulting solution. This reduces the computational complexity of the problem. There is a direct connection between the magnetic flux density and the electrical potential in conductors such as the one we explore, hence a coupling of the sEEG, EEG and MEG data for solving the respective inverse source recovery problems simultaneously is direct. We treat these problems in a unified approach that uses only single and/or double layer potentials. We provide numerical examples using realistic meshes of the head with synthetic data.

The Markov gap for geometric reflected entropy

The reflected entropy SR(A : B) of a density matrix ρAB is a bipartite correlation measure lower-bounded by the quantum mutual information I(A : B). In holographic states satisfying the quantum extremal surface formula, where the reflected entropy is related to the area of the entanglement wedge cross-section, there is often an order-N2 gap between SR and I. We provide an information-theoretic interpretation of this gap by observing that SR− I is related to the fidelity of a particular Markov recovery problem that is impossible in any state whose entanglement wedge cross-section has a nonempty boundary; for this reason, we call the quantity SR− I the Markov gap. We then prove that for time-symmetric states in pure AdS3 gravity, the Markov gap is universally lower bounded by log(2)ℓAdS/2GN times the number of endpoints of the cross-section. We provide evidence that this lower bound continues to hold in the presence of bulk matter, and comment on how it might generalize above three bulk dimensions. Finally, we explore the Markov recovery problem controlling SR− I using fixed area states. This analysis involves deriving a formula for the quantum fidelity — in fact, for all the sandwiched Rényi relative entropies — between fixed area states with one versus two fixed areas, which may be of independent interest. We discuss, throughout the paper, connections to the general theory of multipartite entanglement in holography.

Pauli error estimation via Population Recovery

Motivated by estimation of quantum noise models, we study the problem of learning a Pauli channel, or more generally the Pauli error rates of an arbitrary channel. By employing a novel reduction to the "Population Recovery" problem, we give an extremely simple algorithm that learns the Pauli error rates of an n-qubit channel to precision ϵ in ℓ∞ using just O(1/ϵ2)log⁡(n/ϵ) applications of the channel. This is optimal up to the logarithmic factors. Our algorithm uses only unentangled state preparation and measurements, and the post-measurement classical runtime is just an O(1/ϵ) factor larger than the measurement data size. It is also impervious to a limited model of measurement noise where heralded measurement failures occur independently with probability ≤1/4.We then consider the case where the noise channel is close to the identity, meaning that the no-error outcome occurs with probability 1−η. In the regime of small η we extend our algorithm to achieve multiplicative precision 1±ϵ (i.e., additive precision ϵη) using just O(1ϵ2η)log⁡(n/ϵ) applications of the channel.

Inferring the Hidden Cascade Infection over Erdös-Rényi (ER) Random Graph

Finding hidden infected nodes is extremely important when information or diseases spread rapidly in a network because hints regarding the global properties of the diffusion dynamics can be provided, and effective control strategies for mitigating such spread can be derived. In this study, to understand the impact of the structure of the underlying network, a cascade infection-recovery problem is considered over an Erdös-Rényi (ER) random graph when a subset of infected nodes is partially observed. The goal is to reconstruct the underlying cascade that is likely to generate these observations. To address this, two algorithms are proposed: (i) a Neighbor-based recovery algorithm (NBRA(α)), where 0≤α≤1 is a control parameter, and (ii) a BFS tree-source-based recovery algorithm (BSRA). The first one simply counts the number of infected neighbors for candidate hidden cascade nodes and computes the possibility of infection from the neighbors by controlling the parameter α. The latter estimates the cascade sources first and computes the infection probability from the sources. A BFS tree approximation is used for the underlying ER random graph with respect to the sources for computing the infection probability because of the computational complexity in general loopy graphs. We then conducted various simulations to obtain the recovery performance of the two proposed algorithms. As a result, although the NBRA(α) uses only local information of the neighboring infection status, it recovers the hidden cascade infection well and is not significantly affected by the average degree of the ER random graph, whereas the BSRA works well on a local tree-like structure.

Low-rank matrix recovery with Ky Fan 2-k-norm

Low-rank matrix recovery problem is difficult due to its non-convex properties and it is usually solved using convex relaxation approaches. In this paper, we formulate the non-convex low-rank matrix recovery problem exactly using novel Ky Fan 2-k-norm-based models. A general difference of convex functions algorithm (DCA) is developed to solve these models. A proximal point algorithm (PPA) framework is proposed to solve sub-problems within the DCA, which allows us to handle large instances. Numerical results show that the proposed models achieve high recoverability rates as compared to the truncated nuclear norm method and the alternating bilinear optimization approach. The results also demonstrate that the proposed DCA with the PPA framework is efficient in handling larger instances.