Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen.

By J. Schur

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Pure Mathematics

Citations in the corpus

The Surprising Simplicity of the Early-Time Learning Dynamics of Neural Networks (2020, neutral): “3 (Schur [1911]).”
Generalization Guarantees for Neural Networks via Harnessing the Low-rank Structure of the Jacobian (2019, methodology): “Next we state a useful lemma from [52] which allows us to bound the eigenvalues of the Hadamard product of the two PSD matrices.”
Best constants for certain multilinear integral operators (2006, methodology): “This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.”
Toward Moderate Overparameterization: Global Convergence Guarantees for Training Shallow Neural Networks (2019, background): “We begin this section by stating a key lemma regarding the spectrum of the Hadamard product of matrices due to Schur [49] which plays a crucial role in both the upper and lower bounds on the eigenvalues of the Jacobian discussed in this section as well as our results on the perturbation of eigenvalu”
Fast Convergence of Natural Gradient Descent for Overparameterized Neural Networks (2019, background): “[Schur [1911]] For two positive deﬁnite matrices A and B , we have The diagonal entries of XX (cid:62) are all 1 since the inputs are normalized.”
Spectra of digraphs (2010, background): “It follows from an inequality of Schur [97] that”
Support Theory for Preconditioning (2003, unknown): “Schur [25] proved several properties of the Hadamard product, including the important results below.”
Regularized estimation of large covariance matrices (2008, background): “This fact was proved by Schur [30] and is also easily seen via a probabilistic interpretation: if X and Y are independent, mean 0 random vectors with Cov(X) = A, Cov(Y) = B, then Cov(X ∗Y) = A ∗B.”
Individualness and Determinantal Point Processes for Pedestrian Detection (2016, background): “First, from the Schur-product theorem [20], S = S◦S is a positive semi-definite matrix where ◦ is a Hadamard product or an element-wise product of matrices.”
The Theory of Quantum Information (2018, background): “Schur [190] proved that the positive semidefinite cone is closed under entry-wise products—which is a fact now referred to as the Schur product theorem.”
Rank-Sparsity Incoherence for Matrix Decomposition (2009, result): “For any matrix M , we have from the results in [28] that”
Sharp bounds for m-linear Hardy and Hilbert Operators (2012, unknown): “0 T (1)(f)(x)g(x) dx ≤ π sin(π/p) ‖f‖Lp(0,∞)‖g‖Lp′ (0,∞), where 1 p + 1 p′ = 1, 1 < p <∞; see for instance [13], [10], and [6].”