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The traces used in side-channel analysis are essential to breaking the key of encryption and the signal quality greatly affects the correct rate of key guessing. Therefore, the preprocessing of side-channel traces plays an important role in side-channel analysis. The process of side-channel leakage signal acquisition is usually affected by internal circuit noise, external environmental noise, and other factors, so the collected signal is often mixed with strong noise. In order to extract the feature information of side-channel signals from very low signal-to-noise ratio traces, a hybrid threshold denoising framework using singular value decomposition is proposed for side-channel analysis preprocessing. This framework is based on singular value decomposition and introduces low-rank matrix approximation theory to improve the rank selection methods of singular value decomposition. This paper combines the hard threshold method of truncated singular value decomposition with the soft threshold method of singular value shrinkage damping and proposes a hybrid threshold denoising framework using singular value decomposition for the data preprocessing step of side-channel analysis as a general preprocessing method for non-profiled side-channel analysis. The data used in the experimental evaluation are from the raw traces of the public database of DPA contest V2 and AES_HD. The success rate curve of non-profiled side-channel analysis further confirms the effectiveness of the proposed framework. Moreover, the signal-to-noise ratio of traces is significantly improved after preprocessing, and the correlation with the correct key is also significantly enhanced. Experimental results on DPA v2 and AES_HD show that the proposed noise reduction framework can be effectively applied to the side-channel analysis preprocessing step, and can successfully improve the signal-to-noise ratio of the traces and the attack efficiency. [ABSTRACT FROM AUTHOR]

We give a Coulson integral formula and a Coulson-Jacobs formula for the p-Schatten energy of bipartite graphs. We use these formulas to compare the p-Schatten energy of different trees by using a quasiorder, establishing the maximality of paths and minimality of stars among all trees. [ABSTRACT FROM AUTHOR]

We develop heuristic interpolation methods for the functions t ↦ log det A + t B and t ↦ trace (A + t B) p where the matrices A and B are Hermitian and positive (semi) definite and p and t are real variables. These functions are featured in many applications in statistics, machine learning, and computational physics. The presented interpolation functions are based on the modification of sharp bounds for these functions. We demonstrate the accuracy and performance of the proposed method with numerical examples, namely, the marginal maximum likelihood estimation for Gaussian process regression and the estimation of the regularization parameter of ridge regression with the generalized cross-validation method. [ABSTRACT FROM AUTHOR]

The traces used in side-channel analysis are essential to breaking the key of encryption and the signal quality greatly affects the correct rate of key guessing. Therefore, the preprocessing of side-channel traces plays an important role in side-channel analysis. The process of side-channel leakage signal acquisition is usually affected by internal circuit noise, external environmental noise, and other factors, so the collected signal is often mixed with strong noise. In order to extract the feature information of side-channel signals from very low signal-to-noise ratio traces, a hybrid threshold denoising framework using singular value decomposition is proposed for side-channel analysis preprocessing. This framework is based on singular value decomposition and introduces low-rank matrix approximation theory to improve the rank selection methods of singular value decomposition. This paper combines the hard threshold method of truncated singular value decomposition with the soft threshold method of singular value shrinkage damping and proposes a hybrid threshold denoising framework using singular value decomposition for the data preprocessing step of side-channel analysis as a general preprocessing method for non-profiled side-channel analysis. The data used in the experimental evaluation are from the raw traces of the public database of DPA contest V2 and AES_HD. The success rate curve of non-profiled side-channel analysis further confirms the effectiveness of the proposed framework. Moreover, the signal-to-noise ratio of traces is significantly improved after preprocessing, and the correlation with the correct key is also significantly enhanced. Experimental results on DPA v2 and AES_HD show that the proposed noise reduction framework can be effectively applied to the side-channel analysis preprocessing step, and can successfully improve the signal-to-noise ratio of the traces and the attack efficiency.

For high‐dimensional data, the standard empirical estimator for the covariance matrix is very poor, and thus many methods have been proposed to more accurately estimate the covariance structure of high‐dimensional data. In this article, we consider estimation under the assumption of sparsity but regularize with respect to the individual false‐positive rate for incorrectly including a matrix entry in the support of the final estimator. The two benefits of this approach are (1) an interpretable regularization parameter removing the need for computationally expensive tuning and (2) extremely fast computation time arising from use of a binary search algorithm implemented to find the best estimator within a carefully constructed operator norm ball. We compare our approach to universal thresholding estimators and lasso‐style penalized estimators on both simulated data and data from gene expression for cancerous tumours. [ABSTRACT FROM AUTHOR]

We define a p-norm in the context of quantum random variables, measurable operator-valued functions with respect to a positive operator-valued measure. This norm leads to a operator-valued L p space that is shown to be complete. Various other norm candidates are considered as well as generalizations of Hölder's inequality to this new context. [ABSTRACT FROM AUTHOR]

The general problem under consideration is the determination of the best (i.e., smallest) constant C p , q , r such that ‖ X Y − Y X ‖ p ≤ C p , q , r ‖ X ‖ q ‖ Y ‖ r for all n × n matrices X and Y , where ‖ ⋅ ‖ p is the Schatten p -norm. Among the open situations, the problem is solved when (i) 2 < p < ∞ , q = r = 1 ; (ii) 2 < p < ∞ , q = 1 , 1 < r < 2 , and X , Y are 2 × 2 real matrices. The result for (i) is obtained via the study of the region (recently obtained in [5]) of singular values of the commutator X Y − Y X , where X and Y are normalized rank one matrices. As a result of independent interest, for 2 × 2 real matrices, the region of singular values of the commutator X Y − Y X , where X is rank one and normalized, and ‖ Y ‖ r = 1 , is determined. The result for (ii) is a consequence of the case when r lies between 1 and 2. [ABSTRACT FROM AUTHOR]

In this paper, we fully characterize the duality mapping over the space of matrices that are equipped with Schatten norms. Our approach is based on the analysis of the saturation of the Hölder inequality for Schatten norms. We prove in our main result that, for p ∈ (1 , ∞) , the duality mapping over the space of real-valued matrices with Schatten-p norm is a continuous and single-valued function and provide an explicit form for its computation. For the special case p = 1, the mapping is set-valued; by adding a rank constraint, we show that it can be reduced to a Borel-measurable single-valued function for which we also provide a closed-form expression. [ABSTRACT FROM AUTHOR]

This paper presents a systematic study of the space complexity of estimating the Schatten p-norms of an n x n matrix in the turnstile streaming model. Both kinds of space complexities, bit complexity and sketching dimension, are considered. Furthermore, two sketching models, general linear sketching and bilinear sketching, are considered. When p is not an even integer, we show that any one-pass algorithm with constant success probability requires near-linear space in terms of bits. This lower bound holds even for sparse matrices, i.e., matrices with O(1) nonzero entries per row and per column. However, when p is an even integer, we give for sparse matrices an upper bound which, up to logarithmic factors, is the same as estimating the pth moment of an n-dimensional vector. These results considerably strengthen lower bounds in previous work for arbitrary (not necessarily sparse) matrices. Similar near-linear lower bounds are obtained for Ky Fan norms, SVD entropy, eigenvalue shrinkers, and M-estimators, many of which could have been solvable in logarithmic space prior to this work. The results for general linear sketches give separations in the sketching complexity of Schatten p-norms with the corresponding vector p-norms, and rule out a table-lookup nearest-neighbor search for p = 1, making progress on a question of Andoni. The results for bilinear sketches are tight for the rank problem and nearly tight for p ≥2; the latter is the first general subquadratic upper bound for sketching the Schatten norms. [ABSTRACT FROM AUTHOR]

In this paper we consider a class of weighted composition operators Rα; a ∈ D defined on the Bergman space Lα²(U+) of the upper half plane. We showed that these classes of operators are unitary, selfadjoint and have numerical radius 1: We calculated the fixed points of these unitary operators and characterized the reducing subspace of T ∈ L(La²(U+)) that commutes with Ra: We also derived various algebraic properties of bounded linear operators defined on La²(U+); in terms of certain distance estimates involving the weighted composition operators Ra: Our main focus is on Toeplitz operators defined on La²(U+). [ABSTRACT FROM AUTHOR]

The Thompson metric provides key geometric insights in the study of non-linear matrix equations and in many optimization problems. However, knowing that an approximate solution is within d T units, in the Thompson metric, of the actual solution provides little insight into how good the approximation is as a matrix or vector approximation. That is, bounding the Thompson metric between an approximate and accurate solution to a problem does not provide obvious bounds either for the spectral or the Frobenius norm, both Schatten norms, of the difference between the approximation and accurate solution. This paper reports such an upper bound, namely that ∥ X − Y ∥ p ≤ 2 1 p e d − 1 e d max ∥ X ∥ p , ∥ Y ∥ p where ⋅ p denotes the Schatten p-norm and d denotes the Thompson metric between X and Y. Furthermore, a more geometric proof leads to a slightly better bound in the case of the Frobenius norm, ∥ X − Y ∥ 2 ≤ e d − 1 e 2 d + 1 ∥ X ∥ 2 2 + ∥ Y ∥ 2 2 ≤ 2 1 2 e d − 1 e 2 d + 1 max ∥ X ∥ p , ∥ Y ∥ p . [ABSTRACT FROM AUTHOR]

Inaccurate knowledge of channel state information (CSI) limits the performance offered by multiple-input multiple-output (MIMO) communications. The design of a MIMO precoder needs to take the CSI uncertainty into consideration to mitigate its effects. This paper proposes a method to design the precoder for a secondary user in an underlay cognitive radio framework that maximizes its transmission performance, where the direct link to its receiver and the interference link to a primary user have CSI uncertainties. We model the CSI uncertainties through the Schatten norm that encompasses most commonly used norm measures such as the spectral and Frobenious norms. The proposed method solves iteratively a minimax problem by deriving the optimal solution for the worst case interference CSI uncertainty, applying the alternate-iterate technique for the worst case direct link CSI deviation, and developing a feasible direction projected subgradient technique for the precoder. Simpler solutions for the precoder are also derived under some specific norm measure of CSI uncertainty and certain requirement of transmission power. Simulations corroborate the expected performance of the proposed design. [ABSTRACT FROM PUBLISHER]

In 1978 Gutman introduced the energy of a graph as the sum of the absolute values of graph eigenvalues, and ever since then graph energy has been intensively studied. Since graph energy is the trace norm of the adjacency matrix, matrix norms provide a natural background for its study. Thus, this paper surveys research on matrix norms that aims to expand and advance the study of graph energy. The focus is exclusively on the Ky Fan and the Schatten norms, both generalizing and enriching the trace norm. As it turns out, the study of extremal properties of these norms leads to numerous analytic problems with deep roots in combinatorics. The survey brings to the fore the exceptional role of Hadamard matrices, conference matrices, and conference graphs in matrix norms. In addition, a vast new matrix class is studied, a relaxation of symmetric Hadamard matrices. The survey presents solutions to just a fraction of a larger body of similar problems bonding analysis to combinatorics. Thus, open problems and questions are raised to outline topics for further investigation. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of precoder design that maximizes the throughput of a secondary user (SU) in an underlay multiple-input multiple-output cognitive radio network, where the channel state information (CSI) from the SU to the primary user (PU) is unavailable or inaccurate. The design maintains the quality of service for the PU through an interference amount measure in terms of the interference temperature or the leakage rate. For the case of unknown CSI, we propose an iterative adaptation algorithm by exploiting the side information in the primary communication network. For the case of imperfect CSI, we model the amount of uncertainty to be within a convex set defined by the Schatten norm and apply the maximin optimization to obtain the solution. To complete this paper, we derive the conditions on the CSI uncertainty radius under which the robust design with imperfect CSI would not perform better than the one with unknown CSI, when using the interference temperature metric. The proposed techniques are supported by numerical simulations. [ABSTRACT FROM PUBLISHER]

Properties of the trace norm geometric discord of the system of two qutrits are studied. The geometric discord of qutrit Bell states, Werner states and bound entangled states is computed. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: In this note, some norm inequalities for the commutator and for the expression are given. [Copyright &y& Elsevier]

Abstract: Murthy and Sethi [M.N. Murthy, V.K. Sethi, Sankhya Ser. B 27 (1965) 201–210] gave a sharp upper bound on the variance of a real random variable in terms of the range of values of that variable. We generalise this bound to the complex case and, more importantly, to the matrix case. In doing so, we make contact with several geometrical and matrix analytical concepts, such as the numerical range, and introduce the new concept of radius of a matrix. We also give a new and simplified proof for a sharp upper bound on the Frobenius norm of commutators recently proven by Böttcher and Wenzel [A. Böttcher, D. Wenzel, The Frobenius norm and the commutator, Linear Algebra Appl. 429 (2008) 1864–1885] and point out that at the heart of this proof lies exactly the matrix version of the variance we have introduced. As an immediate application of our variance bounds we obtain stronger versions of Böttcher and Wenzel’s upper bound. [Copyright &y& Elsevier]

Abstract: King and Ruskai asked whether the norm of a completely positive map , acting between Schatten p and q classes of Hermitian operators,is equal to the norm of that map when acting between Schatten classes of general, not necessarily Hermitian, operators. The first proof of this statement has been given by Watrous. Here, we give an alternative proof that is also valid for 2-positive maps . [Copyright &y& Elsevier]

In [8] , an exact integral formula was proved for the Schatten p -norm of Hankel operators on the one-variable Hardy space for p = 2 , 4, and 6. Moreover, it was shown that these are the only values of p for which such a formula is possible. It was further pointed out in [8] that the only possibility for a meaningful multi-variable analogue of this formula is the case where n = 2 and p = 6. We show that the conjectured integral formula indeed holds in this one possible multi-variable case. [ABSTRACT FROM AUTHOR]

Abstract: We conjecture the following so-called norm compression inequality for partitioned block matrices and the Schatten p-norms: for ,while for the ordering of the inequality is reversed. This inequality includes Hanner’s inequality for matrices as a special case. We prove several special cases of this inequality and show that the partitioning in blocks is essential, by exhibiting counterexamples in the case of and larger partitionings. [Copyright &y& Elsevier]

We provide upper bounds for unitarily invariant norms of finite Toeplitz matrices generated by functions with a Fisher-Hartwig singularity. These bounds are sharp as the matrix size goes to infinity. Our results improve previous estimates and confirm a conjecture recently raised by Böttcher. [ABSTRACT FROM AUTHOR]

Abstract: Let A be a positive semidefinite matrix, block partitioned aswhere B and D are square blocks. We prove the following inequalities for the Schatten q-norm ∥·∥ q :andWe show that these bounds obey a strong sharpness condition when the blocks are of size at least 2×2, and ∥B∥ q , ∥D∥ q ⩾∥C∥ q . Finally, our bounds can be extended to symmetric partitionings into larger numbers of blocks: for A =[A ij ],while for 2⩽ q the inequality is reversed. [Copyright &y& Elsevier]