Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A coordinate-free condition number for convex programming   We introduce and analyze a natural geometric version of Renegar's condition
number R for the homogeneous convex feasibility problem associated with a
regular cone C subseteq R^n. Let Gr_{n,m} denote the Grassmann manifold of
m-dimensional linear subspaces of R^n and consider the projection distance
d_p(W_1,W_2) := ||Pi_{W_1} - Pi_{W_2}|| (spectral norm) between W_1 and W_2 in
Gr_{n,m}, where Pi_{W_i} denotes the orthogonal projection onto W_i. We call
C_G(W) := max {d_p(W,W')^{-1} | W' \in Sigma_m} the Grassmann condition number
of W in Gr_{n,m}, where the set of ill-posed instances Sigma_m subset Gr_{n,m}
is defined as the set of linear subspaces touching C. We show that if W =
im(A^T) for a matrix A in R^{m\times n}, then C_G(W) \le R(A) \le C_G(W)
kappa(A), where kappa(A) =||A|| ||A^\dagger|| denotes the matrix condition
number. This extends work by Belloni and Freund in Math. Program. 119:95-107
(2009). Furthermore, we show that C_G(W) can as well be characterized in terms
of the Riemannian distance metric on Gr_{n,m}. This differential geometric
characterization of C_G(W) is the starting point of the sequel
[arXiv:1112.2603] to this paper, where the first probabilistic analysis of
Renegar's condition number for an arbitrary regular cone C is achieved.

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The categories of this paper are: patt-sol
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The categories of this paper are: dg-ga
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The categories of this paper are: q-fin
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The categories of this paper are: physics
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The categories of this paper are: hep-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: chem-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: nucl-th
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The categories of this paper are: cs
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The categories of this paper are: atom-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-th
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The categories of this paper are: mtrl-th
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The categories of this paper are: acc-phys
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The categories of this paper are: cond-mat