A Survey of the Hadamard Maximal Determinant Problem

In a celebrated paper of 1893, Hadamard established the maximal determinant theorem, which establishes an upper bound on the determinant of a matrix with complex entries of norm at most 1. His paper concludes with the suggestion that mathematicians study the maximum value of the determinant of an $n \times n$ matrix with entries in $\{ \pm 1\}$. This is the Hadamard maximal determinant problem. This survey provides complete proofs of the major results obtained thus far. We focus equally on upper bounds for the determinant (achieved largely via the study of the Gram matrices), and constructive lower bounds (achieved largely via quadratic residues in finite fields and concepts from design theory). To provide an impression of the historical development of the subject, we have attempted to modernise many of the original proofs, while maintaining the underlying ideas. Thus some of the proofs have the flavour of determinant theory, and some appear in print in English for the first time. We survey constructions of matrices in order $n \equiv 3 \mod 4$, giving asymptotic analysis which has not previously appeared in the literature. We prove that there exists an infinite family of matrices achieving at least 0.48 of the maximal determinant bound. Previously the best known constant for a result of this type was 0.34.

Hilbert–Schmidt frames and their duals

The concept of Hilbert–Schmidt frame (HS-frame) was first introduced by Sadeghi and Arefijamaal in 2012. It is more general than [Formula: see text]-frames, and thus, covers many generalizations of frames. This paper addresses the theory of HS-frames. We present a parametric and algebraic formula for all duals of an arbitrarily given HS-frame; prove that the canonical HS-dual induces a minimal-norm expression of the elements in Hilbert spaces; characterize when an HS-frame is an HS-Riesz basis, and when an HS-Bessel sequence is an HS-Riesz sequence (HS-Riesz basis) in terms of Gram matrices.

The varieties of minimal tomographically complete measurements

Minimal Informationally Complete quantum measurements, or MICs, illuminate the structure of quantum theory and how it departs from the classical. Central to this capacity is their role as tomographically complete measurements with the fewest possible number of outcomes for a given finite dimension. Despite their advantages, little is known about them. We establish general properties of MICs, explore constructions of several classes of them, and make some developments to the theory of MIC Gram matrices. These Gram matrices turn out to be a rich subject of inquiry, relating linear algebra, number theory and probability. Among our results are some equivalent conditions for unbiased MICs, a characterization of rank-1 MICs through the Hadamard product, several ways in which immediate properties of MICs capture the abandonment of classical phase space intuitions, and a numerical study of MIC Gram matrix spectra. We also present, to our knowledge, the first example of an unbiased rank-1 MIC which is not group covariant. This work provides further context to the discovery that the symmetric informationally complete quantum measurements (SICs) are in many ways optimal among MICs. In a deep sense, the ideal measurements of quantum physics are not orthogonal bases.

Rational Criteria for Diagonalizability of Real Matrices

The purpose of this note is to obtain rational criteria for diagonalizability of real matrices through the analysis of the moment and Gram matrices associated to a given real matrix. These concepts were introduced by Horn and Lopatin in [R.A. Horn and A.K. Lopatin. The moment and Gram matrices, distinct eigenvalues and zeroes, and rational criteria for diagonalizability. Linear Algebra and its Applications, 299:153-163, 1999] for complex matrices. However, when the matrix is real, it is possible to combine their results with the Borchardt-Jacobi Theorem, in order to get new and noteworthy rational criteria.

A Feature Space Constraint-Based Method for Change Detection in Heterogeneous Images

With the development of remote sensing technologies, change detection in heterogeneous images becomes much more necessary and significant. The main difficulty lies in how to make input heterogeneous images comparable so that the changes can be detected. In this paper, we propose an end-to-end heterogeneous change detection method based on the feature space constraint. First, considering that the input heterogeneous images are in two distinct feature spaces, two encoders with the same structure are used to extract features, respectively. A decoder is used to obtain the change map from the extracted features. Then, the Gram matrices, which include the correlations between features, are calculated to represent different feature spaces, respectively. The squared Euclidean distance between Gram matrices, termed as feature space loss, is used to constrain the extracted features. After that, a combined loss function consisting of the binary cross entropy loss and feature space loss is designed for training the model. Finally, the change detection results between heterogeneous images can be obtained when the model is trained well. The proposed method can constrain the features of two heterogeneous images to the same feature space while keeping their unique features so that the comparability between features can be enhanced and better detection results can be achieved. Experiments on two heterogeneous image datasets consisting of optical and SAR images demonstrate the effectiveness and superiority of the proposed method.