Lipschitz bijections between boolean functions

We answer four questions from a recent paper of Rao and Shinkar [17] on Lipschitz bijections between functions from {0, 1} n to {0, 1}. (1) We show that there is no O(1)-bi-Lipschitz bijection from Dictator to XOR such that each output bit depends on O(1) input bits. (2) We give a construction for a mapping from XOR to Majority which has average stretch $O(\sqrt{n})$ , matching a previously known lower bound. (3) We give a 3-Lipschitz embedding $\phi \colon \{0,1\}^n \to \{0,1\}^{2n+1}$ such that $${\rm{XOR }}(x) = {\rm{ Majority }}(\phi (x))$$ for all $x \in \{0,1\}^n$ . (4) We show that with high probability there is an O(1)-bi-Lipschitz mapping from Dictator to a uniformly random balanced function.

Disjoint superheavy subsets and fragmentation norms

We present a lower bound for a fragmentation norm and construct a bi-Lipschitz embedding [Formula: see text] with respect to the fragmentation norm on the group [Formula: see text] of Hamiltonian diffeomorphisms of a symplectic manifold [Formula: see text]. As an application, we provide an answer to Brandenbursky’s question on fragmentation norms on [Formula: see text], where [Formula: see text] is a closed Riemannian surface of genus [Formula: see text].

METRIC INEQUALITIES

For every $p\in (0,\infty )$ we associate to every metric space $(X,d_{X})$ a numerical invariant $\mathfrak{X}_{p}(X)\in [0,\infty ]$ such that if $\mathfrak{X}_{p}(X)<\infty$ and a metric space $(Y,d_{Y})$ admits a bi-Lipschitz embedding into $X$ then also $\mathfrak{X}_{p}(Y)<\infty$. We prove that if $p,q\in (2,\infty )$ satisfy $q<p$ then $\mathfrak{X}_{p}(L_{p})<\infty$ yet $\mathfrak{X}_{p}(L_{q})=\infty$. Thus, our new bi-Lipschitz invariant certifies that $L_{q}$ does not admit a bi-Lipschitz embedding into $L_{p}$ when $2<q<p<\infty$. This completes the long-standing search for bi-Lipschitz invariants that serve as an obstruction to the embeddability of $L_{p}$ spaces into each other, the previously understood cases of which were metric notions of type and cotype, which however fail to certify the nonembeddability of $L_{q}$ into $L_{p}$ when $2<q<p<\infty$. Among the consequences of our results are new quantitative restrictions on the bi-Lipschitz embeddability into $L_{p}$ of snowflakes of $L_{q}$ and integer grids in $\ell _{q}^{n}$, for $2<q<p<\infty$. As a byproduct of our investigations, we also obtain results on the geometry of the Schatten $p$ trace class $S_{p}$ that are new even in the linear setting.

AN ENHANCED LIPSCHITZ EMBEDDING CLASSIFIER FOR MULTI-EMOTION SPEECH ANALYSIS

This paper proposes an Enhanced Lipschitz Embedding based Classifier (ELEC) for the classification of multi-emotions from speech signals. ELEC adopts geodesic distance to preserve the intrinsic geometry at all scales of speech corpus, instead of Euclidean distance. Based on the minimal geodesic distance to vectors of different emotions, ELEC maps the high dimensional feature vectors into a lower space. Through analyzing the class labels of the neighbor training vectors in the compressed low space, ELEC classifies the test data into six archetypal emotional states, i.e. neutral, anger, fear, happiness, sadness and surprise. Experimental results on clear and noisy data set demonstrate that compared with the traditional methods of dimensionality reduction and classification, ELEC achieves 15% improvement on average for speaker-independent emotion recognition and 11% for speaker-dependent.