Local Recoverability Research Articles

Correlation between microstructural inhomogeneity and architectural design in additively manufactured NiTi shape memory alloys

ABSTRACT Additively manufactured Nitinol (NiTi) architectured materials, designed with unit cell architectures, hold promise for customisable applications. However, the common assumption of homogeneity in modeling and additive manufacturing of these architectured materials needs further investigation because geometric-dependent melt pool behaviour results in inhomogeneous microstructure and thermomechanical properties. This study shows that property inhomogeneity at the mesoscale is one reason for pseudo-linear response and partial superelasticity of the fabricated NiTi body-centered cubic (BCC) architectured materials. We modeled using a phenomenological constitutive relation and additively manufactured NiTi architectured materials with varying relative densities. These fabricated samples showed distinct microstructural textures and compositions that affected their local recoverability. The edge effects and laser turn regions were identified as the causes underlying the observed microstructural inhomogeneity. The dimensionless Fourier number is used to describe the transition of printing modes. This study provides valuable information on rigorous experimental/computational consistency in future work.

Algebraic compressed sensing

We introduce the broad subclass of algebraic compressed sensing problems, where structured signals are modeled either explicitly or implicitly via polynomials. This includes, for instance, low-rank matrix and tensor recovery. We employ powerful techniques from algebraic geometry to study well-posedness of sufficiently general compressed sensing problems, including existence, local recoverability, global uniqueness, and local smoothness. Our main results are summarized in thirteen questions and answers in algebraic compressed sensing. Most of our answers concerning the minimum number of required measurements for existence, recoverability, and uniqueness of algebraic compressed sensing problems are optimal and depend only on the dimension of the model.

Locally Recoverable Codes on Surfaces

A linear error correcting code is a subspace of a finite-dimensional space over a finite field with a fixed coordinate system. Such a code is said to be locally recoverable with locality r if, for every coordinate, its value at a codeword can be deduced from the value of (certain) r other coordinates of the codeword. These codes have found many recent applications, e.g., to distributed cloud storage. We will discuss the problem of constructing good locally recoverable codes and present some constructions using algebraic surfaces that improve previous constructions and sometimes provide codes that are optimal in a precise sense. The main conceptual contribution of this paper is to consider surfaces fibered over a curve in such a way that each recovery set is constructed from points in a single fiber. This allows us to use the geometry of the fiber to guarantee the local recoverability and use the global geometry of the surface to get a hold on the standard parameters of our codes. We look in detail at situations where the fibers are rational or elliptic curves and provide many examples applying our methods.

A new sufficient condition for sparse vector recovery via ℓ1 − ℓ2 local minimization

In this paper, we provide a necessary condition and a sufficient condition such that any [Formula: see text]-sparse vector [Formula: see text] can be recovered from [Formula: see text] via [Formula: see text] local minimization. Moreover, we further verify that the sufficient condition is naturally valid when the restricted isometry constant of the measurement matrix [Formula: see text] satisfies [Formula: see text]. Compared with the existing [Formula: see text] local recoverability condition [Formula: see text], this result shows that [Formula: see text] local recoverability contains more measurement matrices.

Cyclic LRC-LCD Codes With Hierarchical Locality

Locally recoverable (LRC) codes and linear complementary dual (LCD) codes are both useful in distributed storage systems. In this letter, we establish a connection between LRC codes and LCD codes. We derive some conditions on the construction of cyclic codes with $(r,\delta)$ locality and codes with hierarchical locality so that they are also LCD codes. Some constructed codes are optimal. These codes may be useful in simultaneously providing local recoverability and security against certain attacks in distributed storage systems.

Quantum scrambling and the growth of mutual information

Quantum information scrambling refers to the loss of local recoverability of quantum information, which has found widespread attention from high energy physics to quantum computing. In the present analysis we propose a possible starting point for the development of a comprehensive framework for the thermodynamics of scrambling. To this end, we prove that the growth of entanglement as quantified by the mutual information is lower bounded by the time-dependent change of out-of-time-ordered correlator. We further show that the rate of increase of the mutual information can be upper bounded by the sum of local entropy productions, and the exchange entropy arising from the flow of information between separate partitions of a quantum system. Our results are illustrated for the ion trap system, that was recently used to verify information scrambling in an experiment, and for the Sachdev–Ye–Kitaev model.

The Fidelity of Recovery Is Multiplicative

Fawzi and Renner [Commun. Math. Phys. 340(2):575, 2015] recently established a lower bound on the conditional quantum mutual information (CQMI) of tripartite quantum states $ABC$ in terms of the fidelity of recovery (FoR), i.e. the maximal fidelity of the state $ABC$ with a state reconstructed from its marginal $BC$ by acting only on the $C$ system. The FoR measures quantum correlations by the local recoverability of global states and has many properties similar to the CQMI. Here we generalize the FoR and show that the resulting measure is multiplicative by utilizing semi-definite programming duality. This allows us to simplify an operational proof by Brandao et al. [Phys. Rev. Lett. 115(5):050501, 2015] of the above-mentioned lower bound that is based on quantum state redistribution. In particular, in contrast to the previous approaches, our proof does not rely on de Finetti reductions.

Bounds on the Size of Locally Recoverable Codes

In a locally recoverable or repairable code, any symbol of a codeword can be recovered by reading only a small (constant) number of other symbols. The notion of local recoverability is important in the area of distributed storage where a most frequent error-event is a single storage node failure (erasure). A common objective is to repair the node by downloading data from as few other storage nodes as possible. In this paper, we bound the minimum distance of a code in terms of its length, size, and locality. Unlike the previous bounds, our bound follows from a significantly simple analysis and depends on the size of the alphabet being used. It turns out that the binary Simplex codes satisfy our bound with equality; hence, the Simplex codes are the first example of an optimal binary locally repairable code family. We also provide achievability results based on random coding and concatenated codes that are numerically verified to be close to our bounds.

Blind Source Separation by Nuclear Norm Minimization and Local Recoverability Analysis

We propose a new blind source separation (BSS) algorithm that is effective when Hankel matrices constructed from individual source signals are near low-rank and satisfy a certain near-orthogonality condition. Source separation is achieved by finding a nonsingular reverse-mixing operation that minimizes nuclear norms of Hankel matrices constructed from estimated source signals. The new formulation results in a non-convex optimization problem involving a reverse-mixing matrix. Preliminary analysis of local recoverability of source signals as well as few numerical simulations are presented in this letter.

The Local Recoverability of Risk Aversion and Intertemporal Substitution

Call an economy revealing if underlying preferences are recoverable from asset prices, specifically from the prices of aggregate equity or a one-period discount bond. In a Lucas [Econometrica46 (1978), 1429-4444] model, with Kreps–Porteus [Econometrica46 1978), 185-200] nonexpected utility and Markov output growth rate process, local recoverability is shown to be a generic property. In this sense it is generically true that asset pricing models based on the more general Kreps–Porteus utility have more explanatory power than the usual expected utility, and both the risk aversion and intertemporal aspects of the Kreps–Porteus utility can be recovered from a single dynamic equilibrium. Journal of Economic Literature Classification Numbers: D11, G12, C60, D50, D80.