Impossibility of dimension reduction in the nuclear norm

Let $\mathsf{S}_1$ (the Schatten--von Neumann trace class) denote the Banach space of all compact linear operators $T:\ell_2\to \ell_2$ whose nuclear norm $\|T\|_{\mathsf{S}_1}=\sum_{j=1}^\infty\sigma_j(T)$ is finite, where $\{\sigma_j(T)\}_{j=1}^\infty$ are the singular values of $T$. We prove that for arbitrarily large $n\in \mathbb{N}$ there exists a subset $\mathcal{C}\subseteq \mathsf{S}_1$ with $|\mathcal{C}|=n$ that cannot be embedded with bi-Lipschitz distortion $O(1)$ into any $n^{o(1)}$-dimensional linear subspace of $\mathsf{S}_1$. $\mathcal{C}$ is not even a $O(1)$-Lipschitz quotient of any subset of any $n^{o(1)}$-dimensional linear subspace of $\mathsf{S}_1$. Thus, $\mathsf{S}_1$ does not admit a dimension reduction result \'a la Johnson and Lindenstrauss (1984), which complements the work of Harrow, Montanaro and Short (2011) on the limitations of quantum dimension reduction under the assumption that the embedding into low dimensions is a quantum channel. Such a statement was previously known with $\mathsf{S}_1$ replaced by the Banach space $\ell_1$ of absolutely summable sequences via the work of Brinkman and Charikar (2003). In fact, the above set $\mathcal{C}$ can be taken to be the same set as the one that Brinkman and Charikar considered, viewed as a collection of diagonal matrices in $\mathsf{S}_1$. The challenge is to demonstrate that $\mathcal{C}$ cannot be faithfully realized in an arbitrary low-dimensional subspace of $\mathsf{S}_1$, while Brinkman and Charikar obtained such an assertion only for subspaces of $\mathsf{S}_1$ that consist of diagonal operators (i.e., subspaces of $\ell_1$). We establish this by proving that the Markov 2-convexity constant of any finite dimensional linear subspace $X$ of $\mathsf{S}_1$ is at most a universal constant multiple of $\sqrt{\log \mathrm{dim}(X)}$

Dimension Reduction Techniques for l_p (1<p<2), with Applications

For Euclidean space (l_2), there exists the powerful dimension reduction transform of Johnson and Lindenstrauss [Conf. in modern analysis and probability, AMS 1984], with a host of known applications. Here, we consider the problem of dimension reduction for all l_p spaces 1<p<2. Although strong lower bounds are known for dimension reduction in l_1, Ostrovsky and Rabani [JACM 2002] successfully circumvented these by presenting an l_1 embedding that maintains fidelity in only a bounded distance range, with applications to clustering and nearest neighbor search. However, their embedding techniques are specific to l_1 and do not naturally extend to other norms. In this paper, we apply a range of advanced techniques and produce bounded range dimension reduction embeddings for all of 1<p<2, thereby demonstrating that the approach initiated by Ostrovsky and Rabani for l_1 can be extended to a much more general framework. We also obtain improved bounds in terms of the intrinsic dimensionality. As a result we achieve improved bounds for proximity problems including snowflake embeddings and clustering

Modular nuclearity: A generally covariant perspective

A quantum field theory in its algebraic description may admit many irregular states. So far, selection criteria to distinguish physically reasonable states have been restricted to free fields (Hadamard condition) or to flat spacetimes (e.g. Buchholz-Wichmann nuclearity). We propose instead to use a modular l^p-condition, which is an extension of a strengthened modular nuclearity condition to generally covariant theories. The modular nuclearity condition was previously introduced in Minkowski space, where it played an important role in constructive two dimensional algebraic QFT's. We show that our generally covariant extension of this condition makes sense for a vast range of theories, and that it behaves well under causal propagation and taking mixtures. In addition we show that our modular l^p-condition holds for every quasi-free Hadamard state of a free scalar quantum field (regardless of mass or scalar curvature coupling). However, our condition is not equivalent to the Hadamard condition.Comment: 42 page

Einstein equations in the null quasi-spherical gauge III: numerical algorithms

We describe numerical techniques used in the construction of our 4th order evolution for the full Einstein equations, and assess the accuracy of representative solutions. The code is based on a null gauge with a quasi-spherical radial coordinate, and simulates the interaction of a single black hole with gravitational radiation. Techniques used include spherical harmonic representations, convolution spline interpolation and filtering, and an RK4 "method of lines" evolution. For sample initial data of "intermediate" size (gravitational field with 19% of the black hole mass), the code is accurate to 1 part in 10^5, until null time z=55 when the coordinate condition breaks down.Comment: Latex, 38 pages, 29 figures (360Kb compressed