181,247 research outputs found
Scaled-free objects II
This work creates two categories of "array-weighted sets" for the purposes of
constructing universal matrix-normed spaces and algebras. These universal
objects have the analogous universal property to the free vector space, lifting
maps completely bounded on a generation set to a completely bounded linear map
of the matrix-normed space.
Moreover, the universal matrix-normed algebra is used to prove the existence
of a free product for matrix-normed algebras using algebraic methods.Comment: 46 pages. Version 4 fixed a few minor typos. Version 3 added
matricial completion; fixed an arithmetic error in Example 3.5.10. Version 2
added a preliminaries section on weighted sets and matricial Banach spaces,
incorporating much of "Matricial Banach spaces" in summary; fixed a domain
issue in Lemma 3.3.2; simplified Examples 3.5.10 and 4.11; added more proofs
to Sections 4 and
Detection thresholds in very sparse matrix completion
Let be a rectangular matrix of size and be the random
matrix where each entry of is multiplied by an independent
-Bernoulli random variable with parameter . This paper is about
when, how and why the non-Hermitian eigen-spectra of the randomly induced
asymmetric matrices and captures more of the
relevant information about the principal component structure of than via
its SVD or the eigen-spectra of and , respectively. Hint: the
asymmetry inducing randomness breaks the echo-chamber effect that cripples the
SVD.
We illustrate the application of this striking phenomenon on the low-rank
matrix completion problem for the setting where each entry is observed with
probability , including the very sparse regime where is of order ,
where matrix completion via the SVD of fails or produces unreliable
recovery. We determine an asymptotically exact, matrix-dependent, non-universal
detection threshold above which reliable, statistically optimal matrix recovery
using a new, universal data-driven matrix-completion algorithm is possible.
Averaging the left and right eigenvectors provably improves the recovered
matrix but not the detection threshold. We define another variant of this
asymmetric procedure that bypasses the randomization step and has a detection
threshold that is smaller by a constant factor but with a computational cost
that is larger by a polynomial factor of the number of observed entries. Both
detection thresholds shatter the seeming barrier due to the well-known
information theoretical limit for matrix completion found in
the literature.Comment: 84 pages, 10 pictures. Submitte
Positive semidefinite matrix completion, universal rigidity and the Strong Arnold Property
This paper addresses the following three topics: positive semidefinite (psd) matrix completions, universal rigidity of frameworks, and the Strong Arnold Property (SAP). We show some strong connections among these topics, using semidefinite programming as unifying theme. Our main contribution is a sufficient condition for constructing partial psd matrices which admit a unique completion to a full psd matrix. Such partial matrices are an essential tool in the study of the Gram dimension \gd(G) of a graph G, a recently studied graph parameter related to the low psd matrix completion problem. Additionally, we derive an elementary proof of Connelly's sufficient condition for universal rigidity of tensegrity frameworks and we investigate the links between these two sufficient conditions. We also give a geometric characterization of psd matrices satisfying the Strong Arnold Property in terms of nondegeneracy of an associated semidefinite program, which we use to establish some links between the Gram dimension \gd(\cdot) and the Colin de Verdi\`ere type graph parameter ν=(⋅)
Positive semidefinite matrix completion, universal rigidity and the Strong Arnold Property
This paper addresses the following three topics: positive semidefinite (psd) matrix completions, universal rigidity of frameworks, and the Strong Arnold Property (SAP). We show some strong connections among these topics, using semidefinite programming as unifying theme. Our main contribution is a sufficient condition for constructing partial psd matrices which admit a unique completion to a full psd matrix. Such partial matrices are an essential tool in the study of the Gram dimension \gd(G) of a graph G, a recently studied graph parameter related to the low psd matrix completion problem. Additionally, we derive an elementary proof of Connelly's sufficient condition for universal rigidity of tensegrity frameworks and we investigate the links between these two sufficient conditions. We also give a geometric characterization of psd matrices satisfying the Strong Arnold Property in terms of nondegeneracy of an associated semidefinite program, which we use to establish some links between the Gram dimension \gd(\cdot) and the Colin de Verdi\`ere type graph parameter ν=(⋅)
Positive semidefinite matrix completion, universal rigidity and the Strong Arnold Property
This paper addresses the following three topics: positive semidefinite (psd) matrix completions, universal rigidity of frameworks, and the Strong Arnold Property (SAP). We show some strong connections among these topics, using semidefinite programming as unifying theme. Our main contribution is a sufficient condition for constructing partial psd matrices which admit a unique completion to a full psd matrix. Such partial matrices are an essential tool in the study of the Gram dimension \gd(G) of a graph G, a recently studied graph parameter related to the low psd matrix completion problem. Additionally, we derive an elementary proof of Connelly's sufficient condition for universal rigidity of tensegrity frameworks and we investigate the links between these two sufficient conditions. We also give a geometric characterization of psd matrices satisfying the Strong Arnold Property in terms of nondegeneracy of an associated semidefinite program, which we use to establish some links between the Gram dimension \gd(\cdot) and the Colin de Verdi\`ere type graph parameter ν=(⋅)
Positive semidefinite matrix completion, universal rigidity and the Strong Arnold Property
This paper addresses the following three topics: positive semidefinite (psd) matrix completions, universal rigidity of frameworks, and the Strong Arnold Property (SAP). We show some strong connections among these topics, using semidefinite programming as unifying theme. Our main contribution is a sufficient condition for constructing partial psd matrices which admit a unique completion to a full psd matrix. Such partial matrices are an essential tool in the study of the Gram dimension \gd(G) of a graph G, a recently studied graph parameter related to the low psd matrix completion problem. Additionally, we derive an elementary proof of Connelly's sufficient condition for universal rigidity of tensegrity frameworks and we investigate the links between these two sufficient conditions. We also give a geometric characterization of psd matrices satisfying the Strong Arnold Property in terms of nondegeneracy of an associated semidefinite program, which we use to establish some links between the Gram dimension \gd(\cdot) and the Colin de Verdi\`ere type graph parameter ν=(⋅)
- …