Chaos, Complexity, and Random Matrices

Chaos and complexity entail an entropic and computational obstruction to describing a system, and thus are intrinsically difficult to characterize. In this paper, we consider time evolution by Gaussian Unitary Ensemble (GUE) Hamiltonians and analytically compute out-of-time-ordered correlation functions (OTOCs) and frame potentials to quantify scrambling, Haar-randomness, and circuit complexity. While our random matrix analysis gives a qualitatively correct prediction of the late-time behavior of chaotic systems, we find unphysical behavior at early times including an $\mathcal{O}(1)$ scrambling time and the apparent breakdown of spatial and temporal locality. The salient feature of GUE Hamiltonians which gives us computational traction is the Haar-invariance of the ensemble, meaning that the ensemble-averaged dynamics look the same in any basis. Motivated by this property of the GUE, we introduce $k$-invariance as a precise definition of what it means for the dynamics of a quantum system to be described by random matrix theory. We envision that the dynamical onset of approximate $k$-invariance will be a useful tool for capturing the transition from early-time chaos, as seen by OTOCs, to late-time chaos, as seen by random matrix theory.Comment: 61 pages, 14 figures; v2: references added, typos fixe

Fast human detection for video event recognition

Human body detection, which has become a research hotspot during the last two years, can be used in many video content analysis applications. This paper investigates a fast human detection method for volume based video event detection. Compared with other object detection systems, human body detection brings more challenge due to threshold problems coming from a wide range of dynamic properties. Motivated by approaches successfully introduced in facial recognition applications, it adapts and adopts feature extraction and machine learning mechanism to classify certain areas from video frames. This method starts from the extraction of Haar-like features from large numbers of sample images for well-regulated feature distribution and is followed by AdaBoost learning and detection algorithm for pattern classification. Experiment on the classifier proves the Haar-like feature based machine learning mechanism can provide a fast and steady result for human body detection and can be further applied to reduce negative aspects in human modelling and analysis for volume based event detection

Integration and measures on the space of countable labelled graphs

In this paper we develop a rigorous foundation for the study of integration and measures on the space $\mathscr{G}(V)$ of all graphs defined on a countable labelled vertex set $V$. We first study several interrelated $\sigma$-algebras and a large family of probability measures on graph space. We then focus on a "dyadic" Hamming distance function $\left\| \cdot \right\|_{\psi,2}$, which was very useful in the study of differentiation on $\mathscr{G}(V)$. The function $\left\| \cdot \right\|_{\psi,2}$ is shown to be a Haar measure-preserving bijection from the subset of infinite graphs to the circle (with the Haar/Lebesgue measure), thereby naturally identifying the two spaces. As a consequence, we establish a "change of variables" formula that enables the transfer of the Riemann-Lebesgue theory on $\mathbb{R}$ to graph space $\mathscr{G}(V)$. This also complements previous work in which a theory of Newton-Leibnitz differentiation was transferred from the real line to $\mathscr{G}(V)$ for countable $V$. Finally, we identify the Pontryagin dual of $\mathscr{G}(V)$, and characterize the positive definite functions on $\mathscr{G}(V)$.Comment: 15 pages, LaTe