Matrix analysis

By Roger A. Horn, Charles R. Johnson. 1985

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In Pure Mathematics

Citations in the corpus

On the Equivalence between Kernel Quadrature Rules and Random Feature Expansions (2015, background): “lead to simple behaviors for the spectrum. For example, if we have a second distribution so that dρ′ dρ is upper-bounded by a constantc, then, as a consequence of the Courant-Fischer minimax theorem (Horn and Johnson, 2012), the eigenvalues for dρ′ are less than than ctimes that the ones for dρ. Sim”
Continual Learning with Recursive Gradient Optimization (2022, methodology): “…problem 9 can be expressed as: P : { min P σ̂m(V TΛPV U TΛH̄U) s.t. trace(P ) = dim(P ) (24) As orthogonal transformation maintains eigen value (Horn & Johnson, 2012), which means: σ̂m(V TΛPV U TΛH̄U) = σ̂m(UV TΛPV U TΛH̄) There are two main variable to be optimized, diagonal matrix ΛP…”
DeltaGrad: Rapid retraining of machine learning models (2020, methodology): “Effcient model updating for adding and removing data-points is possible for linear models, based on effcient rank one updates of matrix inverses (e.g., Birattari et al., 1999; Horn & Johnson, 2012; Cao & Yang, 2015, etc).”
Divide and conquer kernel ridge regression: a distributed algorithm with minimax optimal rates (2013, neutral): “Then the matrices Ai = A T i ∈ R2d×2d, and moreover λmax(Ai) = |||Ai||| = ∣∣∣Q−1πiπT i − I ∣∣∣, and similarly for their averages [5].”
Community detection in graphs (2009, background): “The scenario we have described is expected from perturbation theory (Bhatia, 1997; Stewart and Sun, 1990).”
Regularization Techniques for Learning with Matrices (2009, background): “Norms satisfying it are called quadratic symmetric gauge functions (or Q-norms) [Bhatia, 1997, p. 89].”
Gradient Descent with Identity Initialization Efficiently Learns Positive-Definite Linear Transformations by Deep Residual Networks (2018, neutral): “Lemma 17 ((Horn & Johnson, 2013)) .”
The Johnson-Lindenstrauss Transform Itself Preserves Differential Privacy (2012, background): “As mentioned, we focus only on x ∈ V = 1 ⊥ , where As noted above, we have that for every x it holds that It follows that for every x we have that (cid:16) (cid:17) (cid:16) Q and recalling that i ∆ i ≤ 2 and that ∀ i, σ 2 i ≥ w it holds that We now turn to the lower bound of (2).”
Entity Resolution and Federated Learning get a Federated Resolution (2018, background): “Following [5], for any symmetric matrix M, we let λ(M) (resp.”
Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions (2009, neutral): “Standard references for this material include [11, 72].”
High-dimensional Ising model selection using ℓ1-regularized logistic regression (2010, methodology): “Note that the precise value and meaning of K may diﬀer from line to line. 18 P. RAVIKUMAR, M. J. WAINWRIGHT AND J. D. LAFFERTY 5.1. Proof of Lemma 5. By the Courant–Fischer variational representation [16], we have Λmin(QSS) = min kxk2=1 xTQ SSx = min kxk2=1 {xT Qn SSx+x T(Q SS −Q n SS)x} ≤yTQn SSy +”