Conditioning Gaussian measure on Hilbert space

For a Gaussian measure on a separable Hilbert space with covariance operator $C$, we show that the family of conditional measures associated with conditioning on a closed subspace $S^{\perp}$ are Gaussian with covariance operator the short $\mathcal{S}(C)$ of the operator $C$ to $S$. We provide two proofs. The first uses the theory of Gaussian Hilbert spaces and a characterization of the shorted operator by Andersen and Trapp. The second uses recent developments by Corach, Maestripieri and Stojanoff on the relationship between the shorted operator and $C$-symmetric oblique projections onto $S^{\perp}$. To obtain the assertion when such projections do not exist, we develop an approximation result for the shorted operator by showing, for any positive operator $A$, how to construct a sequence of approximating operators $A^{n}$ which possess $A^{n}$-symmetric oblique projections onto $S^{\perp}$ such that the sequence of shorted operators $\mathcal{S}(A^{n})$ converges to $\mathcal{S}(A)$ in the weak operator topology. This result combined with the martingale convergence of random variables associated with the corresponding approximations $C^{n}$ establishes the main assertion in general. Moreover, it in turn strengthens the approximation theorem for shorted operator when the operator is trace class; then the sequence of shorted operators $\mathcal{S}(A^{n})$ converges to $\mathcal{S}(A)$ in trace norm

Invariant types in NIP theories

We study invariant types in NIP theories. Amongst other things: we prove a definable version of the (p,q)-theorem in theories of small or medium directionality; we construct a canonical retraction from the space of M-invariant types to that of M-finitely satisfiable types; we show some amalgamation results for invariant types and list a number of open questions.Comment: Small changes mad

The prospects for mathematical logic in the twenty-first century

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.Comment: Association for Symbolic Logi

Eberlein oligomorphic groups

We study the Fourier--Stieltjes algebra of Roelcke precompact, non-archimedean, Polish groups and give a model-theoretic description of the Hilbert compactification of these groups. We characterize the family of such groups whose Fourier--Stieltjes algebra is dense in the algebra of weakly almost periodic functions: those are exactly the automorphism groups of $\aleph_0$-stable, $\aleph_0$-categorical structures. This analysis is then extended to all semitopological semigroup compactifications $S$ of such a group: $S$ is Hilbert-representable if and only if it is an inverse semigroup. We also show that every factor of the Hilbert compactification is Hilbert-representable.Comment: 23 page

Stability and stable groups in continuous logic

We develop several aspects of local and global stability in continuous first order logic. In particular, we study type-definable groups and genericity

On the matrix square root via geometric optimization

This paper is triggered by the preprint "\emph{Computing Matrix Squareroot via Non Convex Local Search}" by Jain et al. (\textit{\textcolor{blue}{arXiv:1507.05854}}), which analyzes gradient-descent for computing the square root of a positive definite matrix. Contrary to claims of~\citet{jain2015}, our experiments reveal that Newton-like methods compute matrix square roots rapidly and reliably, even for highly ill-conditioned matrices and without requiring commutativity. We observe that gradient-descent converges very slowly primarily due to tiny step-sizes and ill-conditioning. We derive an alternative first-order method based on geodesic convexity: our method admits a transparent convergence analysis ($< 1$ page), attains linear rate, and displays reliable convergence even for rank deficient problems. Though superior to gradient-descent, ultimately our method is also outperformed by a well-known scaled Newton method. Nevertheless, the primary value of our work is its conceptual value: it shows that for deriving gradient based methods for the matrix square root, \emph{the manifold geometric view of positive definite matrices can be much more advantageous than the Euclidean view}.Comment: 8 pages, 12 plots, this version contains several more references and more words about the rank-deficient cas