Trace Norm Research Articles

Revisiting trace norm minimization for tensor Tucker completion: A direct multilinear rank learning approach

To efficiently express tensor data using the Tucker format, a critical task is to minimize the multilinear rank such that the model would not be over-flexible and lead to overfitting. Due to the lack of rank minimization tools in tensor, existing works connect Tucker multilinear rank minimization to trace norm minimization of matrices unfolded from the tensor data. While these formulations try to exploit the common aim of identifying the low-dimensional structure of the tensor and matrix, this paper reveals that existing trace norm-based formulations in Tucker completion are inefficient in multilinear rank minimization. We further propose a new interpretation of Tucker format such that trace norm minimization is applied to the factor matrices of the equivalent representation, rather than some matrices unfolded from tensor data. Based on the newly established problem formulation, a fixed point iteration algorithm is proposed, and its convergence is proved. Numerical results are presented to show that the proposed algorithm exhibits significant improved performance in terms of multilinear rank learning and consequently tensor signal recovery accuracy, compared to existing trace norm based Tucker completion methods.

On the Signless Laplacian ABC-Spectral Properties of a Graph

In the paper, we introduce the signless Laplacian ABC-matrix Q̃(G)=D¯(G)+Ã(G), where D¯(G) is the diagonal matrix of ABC-degrees and Ã(G) is the ABC-matrix of G. The eigenvalues of the matrix Q̃(G) are the signless Laplacian ABC-eigenvalues of G. We give some basic properties of the matrix Q̃(G), which includes relating independence number and clique number with signless Laplacian ABC-eigenvalues. For bipartite graphs, we show that the signless Laplacian ABC-spectrum and the Laplacian ABC-spectrum are the same. We characterize the graphs with exactly two distinct signless Laplacian ABC-eigenvalues. Also, we consider the problem of the characterization of the graphs with exactly three distinct signless Laplacian ABC-eigenvalues and solve it for bipartite graphs and, in some cases, for non-bipartite graphs. We also introduce the concept of the trace norm of the matrix Q̃(G)−tr(Q̃(G))nI, called the signless Laplacian ABC-energy of G. We obtain some upper and lower bounds for signless Laplacian ABC-energy and characterize the extremal graphs attaining it. Further, for graphs of order at most 6, we compare the signless Laplacian energy and the ABC-energy with the signless Laplacian ABC-energy and found that the latter behaves well, as there is a single pair of graphs with the same signless Laplacian ABC-energy unlike the 26 pairs of graphs with same signless Laplacian energy and eight pairs of graphs with the same ABC-energy.

A note on invariant states of Gaussian quantum Markov semigroups

Gaussian quantum Markov semigroup on the algebra of bounded operator on a [Formula: see text]-mode Fock space is characterized either by the property of their predual that preserves quantum Gaussian states. In this paper, show that gaussian QMSs can have only Gaussian invariant states if the spectrum of the drift matrix does not contain points on the imaginary axis. Moreover, in this case, any initial state converges in trace norm toward a unique gaussian invariant state.

Derivation of standard quantum theory via state discrimination

It is a key issue to characterize the model of standard quantum theory out of general models by an operational condition. The framework of general probabilistic theories (GPTs) is a new information theoretical approach to single out standard quantum theory. It is known that traditional properties, for example, Bell-CHSH inequality, are not sufficient to single out standard quantum theory among possible models in GPTs. As a more precise property, we focus on the bound of the performance for an information task called state discrimination in general models. We give an equivalent condition for outperforming the minimum discrimination error probability under the standard quantum theory given by the trace norm. Besides, by applying the equivalent condition, we characterize standard quantum theory out of general models in GPTs by the bound of the performance for state discrimination.

Spectral spread and trace norm of eccentricity matrix of graphs

The spectral spread of the eccentricity matrix E(G) of a graph G is defined as the difference between the largest and the smallest eigenvalue of E(G). While the trace norm of the eccentricity matrix is the absolute sum of the eigenvalues of E(G). In this paper, we obtain various bounds for the spread in terms of the various parameters of G and related it with the trace norm of E(G). We characterize some extremal graphs attaining such bounds.

Tracially amenable actions and purely infinite crossed products

AbstractWe introduce the notion of tracial amenability for actions of discrete groups on unital, tracial C$$^*$$ ∗ -algebras, as a weakening of amenability where all the relevant approximations are done in the uniform trace norm. We characterize tracial amenability with various equivalent conditions, including topological amenability of the induced action on the trace space. Our main result concerns the structure of crossed products: for groups containing the free group $$F_2$$ F 2 , we show that outer, tracially amenable actions on simple, unital, $$\mathcal {Z}$$ Z -stable C$$^*$$ ∗ -algebras always have purely infinite crossed products. Finally, we give concrete examples of tracially amenable actions of free groups on simple, unital AF-algebras.

Enhanced Young-type inequalities utilizing Kantorovich approach for semidefinite matrices

Abstract This article introduces new Young-type inequalities, leveraging the Kantorovich constant, by refining the original inequality. In addition, we present a range of norm-based inequalities applicable to positive semidefinite matrices, such as the Hilbert-Schmidt norm and the trace norm. The importance of these results lies in their dual significance: they hold inherent value on their own, and they also extend and build upon numerous established results within the existing literature.

Typicality in random quantum scattering.

We consider scattering processes where a quantum system comprises an inner subsystem and a boundary and is subject to Haar-averaged random unitaries acting on the boundary-environment Hilbert space only. We show that, regardless of the initial state, a single scattering event will disentangle the unconditional state (i.e., the scattered state when no information about the applied unitary is available) across the inner subsystem-boundary partition. Also, we apply Lévy's lemma to constrain the trace norm fluctuations around the unconditional state. Finally, we derive analytical formulas for the mean scattered purity for initial globally pure states and provide one with numerical evidence of the reduction of fluctuations around such mean values with increasing environmental dimension.

On spectral spread and trace norm of Sombor matrix

On spectral spread and trace norm of Sombor matrix

Uncertainty-Adjusted Recommendation via Matrix Factorization With Weighted Losses.

In a recommender systems (RSs) dataset, observed ratings are subject to unequal amounts of noise. Some users might be consistently more conscientious in choosing the ratings they provide for the content they consume. Some items may be very divisive and elicit highly noisy reviews. In this article, we perform a nuclear-norm-based matrix factorization method which relies on side information in the form of an estimate of the uncertainty of each rating. A rating with a higher uncertainty is considered more likely to be erroneous or subject to large amounts of noise, and therefore more likely to mislead the model. Our uncertainty estimate is used as a weighting factor in the loss we optimize. To maintain the favorable scaling and theoretical guarantees coming with nuclear norm regularization even in this weighted context, we introduce an adjusted version of the trace norm regularizer which takes the weights into account. This regularization strategy is inspired from the weighted trace norm which was introduced to tackle nonuniform sampling regimes in matrix completion. Our method exhibits state-of-the-art performance on both synthetic and real life datasets in terms of various performance measures, confirming that we have successfully used the auxiliary information extracted.

Identification of Low Order Systems in a Loewner Framework

This paper considers the problem of non-parametric identification of low-order models from time-domain experimental data using a combination of Caratheodory Fejer and Loewner-based interpolation, followed by a Loewner matrix Balanced Reduction (LBR) step. As we show in the paper, the Loewner matrix is an estimator for the trace norm of a system, playing a role similar to the one played by the Hankel matrix. However, utilizing Zolotarev numbers to establish decay rate bounds for singular values reveals that the decay of singular values in the Loewner matrix is considerably faster than that in the Hankel matrix. Thus, Loewner-based methods yield lower order systems, with the same error bound, than comparable ones based on Hankel matrices. The effectiveness of our method is demonstrated through a numerical example.

Quantifying channel coherence via the norm distance

Quantifying the number of resources contained in a physical object has been one of the core topics in the resource theory of coherence. In this paper, we introduce the dynamical coherence measures based on a class of norms in the classical channel setting. It is proved that it satisfies faithfulness, decreases monotonically under the maximally incoherent superchannels, and is convex. Moreover, we show that it satisfies subadditivity under both the composition and tensor product of channels. Especially, the diamond measure as a special case is discussed in detail, it can reduce to trace norm of coherence, satisfies amortization inequality, and can be calculated efficiently using a semidefinite program. In addition, we introduce the creation-coherent diamond measure and find that neither the detection coherence nor the creation coherence of a channel exceeds the coherence of the channel, which does not exceed the purity of the channel. Second, we introduce the corresponding dephasing measure, which is a dynamical coherence measure under the dephasing-covariant incoherent superchannels. Meanwhile, we also introduce the dephasing diamond measure as a special case. Third, we use the dephasing diamond measure to accurately calculate the coherence values of some important noisy channels such as amplitude damping channel, phase damping channel, and depolarizing channel, respectively, and give the sufficient and necessary conditions for an unital qubit channel with a parameter probability vector to be a coherent channel. Finally, the operational interpretation of our diamond measure in the binary channel discrimination task is investigated.

Sparse plus low-rank identification for dynamical latent-variable graphical AR models

This paper focuses on the identification of graphical autoregressive models with dynamical latent variables. The dynamical structure of latent variables is described by a matrix polynomial transfer function. Taking account of the sparse interactions between the observed variables and the low-rank property of the latent-variable model, a new sparse plus low-rank optimization problem is formulated to identify the graphical auto-regressive part, which is then handled using the trace approximation and reweighted nuclear norm minimization. Afterwards, the dynamics of latent variables are recovered from low-rank spectral decomposition using the trace norm convex programming method. Simulation examples are used to illustrate the effectiveness of the proposed approach.

Typical bipartite steerability and generalized local quantum measurements

Recently proposed correlation-matrix-based sufficient conditions for bipartite steerability from Alice to Bob for arbitrary measurements are applied to local informationally complete positive operator valued measures (POVMs) of the (N,M)-type. These POVMs include a large class of local generalized measurements of current interest [Siudzińska K 2022 Phys. Rev. A 105 042209]. It is shown that the trace norm of correlation matrices with local (N,M)-POVMs is proportional to that of local orthonormal hermitian operator bases (LOOs). This implies that all types of informationally complete (N,M)-POVMs are equally powerful in detecting bipartite steerability from Alice to Bob and, in addition, they are as powerful as LOOs. In order to explore the typicality of steering numerical calculations of lower bounds on Euclidean volume ratios between steerable bipartite quantum states from Alice to Bob and all quantum states are determined with the help of a hit-and-run Monte-Carlo algorithm. These results demonstrate that with the single exception of two qubits this correlation-matrix-based sufficient condition significantly underestimates these volume ratios. These results are also compared to a recently proposed method that determines bipartite steerability from Alice’s qubit to Bob’s arbitrary dimensional quantum system by bipartite entanglement detection. It is demonstrated that in general this method is significantly more effective in detecting typical steerability provided non local entanglement detection methods are used which transcend local measurements.

Multitask Learning for Classification Problem via New Tight Relaxation of Rank Minimization.

Multitask learning (MTL) is a joint learning paradigm, which fuses multiple related tasks together to achieve the better performance than single-task learning methods. It has been observed by many researchers that different tasks with certain similarities share a low-dimensional common yet latent subspace. In order to get the low-rank structure shared across tasks, trace norm has been used as a convex relaxation of the rank minimization problem. However, trace norm is not a tight approximation for the rank function. To address this important issue, we propose two novel regularization-based models to approximate the rank minimization problem by minimizing the k minimal singular values. For our new models, if the minimal singular values are suppressed to zeros, the rank would also be reduced. Compared with the standard trace norm, our new regularization-based models are the tighter approximations, which can help our models capture the low-dimensional subspace among multiple tasks better. Besides, it is an NP-hard problem to directly solve the exact rank minimization problem for our models. In this article, we proposed two simple but effective strategies to optimize our models, which tactically solves the exact rank minimization problem by setting a large penalizing parameter. Experimental results performed on synthetic and real-world benchmark datasets demonstrate that the proposed models have the ability of learning the low-rank structure shared across tasks and the better performance than other classical MTL methods.

Exponential convergence in the trace norm and time-inhomogeneous Markovian evolutions on ℬ(ℋ)

We consider a sequence of operators [Formula: see text] that describes the time evolution of normal states of an open quantum system. Our objective is to analyze the asymptotic behavior of [Formula: see text], where [Formula: see text] is an arbitrary initial normal state. Our focus is on specific situations in which the operators [Formula: see text] are produced by various quantum Markov semigroups of weak coupling limit-type. To tackle this problem, we introduce a new result concerning exponential convergence to equilibrium in the trace norm for quantum Markov semigroups, provided that their induced semigroups have a spectral gap. Moreover, applications to quantum annealing are discussed.

Quantum MDS codes with new length and large minimum distance

According to the well-known CSS construction, constructing quantum MDS codes are extensively investigated via Hermitian self-orthogonal generalized Reed-Solomon (GRS) codes. In this paper, given two Hermitian self-orthogonal GRS codes GRSk1(A,vA) and GRSk2(B,vB), we propose a sufficient condition to ensure that GRSk(A∪B,vA∪B) is still a Hermitian self-orthogonal code. Consequently, we first present a new general construction of infinitely families of quantum MDS codes from known ones. Moreover, applying the trace function and norm function over finite fields, we give another two new constructions of quantum MDS codes with flexible parameters. It turns out that the forms of the lengths of our quantum MDS codes are quite different from previous known results in the literature. Meanwhile, the minimum distances of all the q-ary quantum MDS codes are bigger than q/2+1.

Relation between the trace norm of an oriented graph and its rank

Let D be an oriented graph, i.e., a digraph in which all arcs are not symmetric, with vertex set {1,…,n}. The adjacency matrix A=(aij) of D is the n×n matrix defined as aij=1 if ij is an arc of D and aij=0 if ij is not an arc of D. The rank of D, written as r(D), is defined to be the rank of its adjacency matrix, and the trace norm of D is defined as N(D)=∑i=1nσi, where σ1≥σ2≥…≥σn≥0 are the singular values of A, i.e., the square roots of the eigenvalues of AAT. In this paper, we establish a lower bound and an upper bound for the trace norm of a connected oriented graph D in terms of its rank and its maximum vertex degree Δ asr(D)≤N(D)≤r(D)Δ, the extremal graphs attaining the bounds are characterized, respectively. Furthermore, we prove that if D is not Pn→ or Cn→, then the lower bound can be improved to r(D)+5−2, where Pn→ and Cn→ respectively denote the orientation of Pn and Cn such that all 2-degree vertices are not sink-source.

Elaborate multi-task subspace learning with discrete group constraint

In multi-task learning (MTL), multiple related tasks can be learned simultaneously under the shared information to improve the generalization performance. However, most of MTL methods assume that all the learning tasks are related indeed and appropriate for joint learning. In some real situations, this assumption may not hold and further lead to the problem of negative transfer. Therefore, in this paper, we not only focus on researching the problem of robustly learning the common feature structure shared by tasks, but also discriminate with which tasks one task should share. By combining with the idea of subspace learning, we propose an elaborate multi-task subspace learning model (EMTSL) with discrete group structure constraint, which can cluster the learned tasks into a set of groups. By introducing the Schatten p-norm instead of trace norm, our model EMTSL can better approximate the low-rank constraint and also avoid the trivial solution. Furthermore, we design an efficient algorithm based on the re-weighted method to solve the proposed model. In addition, the convergence analysis of our algorithm is given in this paper. Experimental results on both synthetic and real-world datasets demonstrate the superiority of our method.

Strategies for single-shot discrimination of process matrices

The topic of causality has recently gained traction quantum information research. This work examines the problem of single-shot discrimination between process matrices which are an universal method defining a causal structure. We provide an exact expression for the optimal probability of correct distinction. In addition, we present an alternative way to achieve this expression by using the convex cone structure theory. We also express the discrimination task as semidefinite programming. Due to that, we have created the SDP calculating the distance between process matrices and we quantify it in terms of the trace norm. As a valuable by-product, the program finds an optimal realization of the discrimination task. We also find two classes of process matrices which can be distinguished perfectly. Our main result, however, is a consideration of the discrimination task for process matrices corresponding to quantum combs. We study which strategy, adaptive or non-signalling, should be used during the discrimination task. We proved that no matter which strategy you choose, the probability of distinguishing two process matrices being a quantum comb is the same.