arXiv subjects

Yang Chu

Publications and source records attributed to Yang Chu.

5 recordsLinked to original sources

arXiv2025

Exponential Absolute Minimizing extension and biased infinity Laplacian

We study the variational structure of the biased infinity Laplacian by introducing a notion of the $\beta$\textit{-Exponential Absolute Minimizing Extension} ($\beta$--AM) on arbitrary length space, which absolutely minimizing the exponential slope $$ L^{\beta}_u (E) := \beta \sup_{x,y \in E} \frac{u(y) - e^{-\beta |x-y|} u(x)}{1- e^{-\beta |x-y|}}. $$We also define the corresponding Exponential McShane-Whitney-type extension, and $\beta$-biased convexity, which equivalently characterize $\beta$-AM and may be of independent interest. These generalize the classical Absolute Minimizing Lipschitz Extension as a special case when $\beta = 0$. In Euclidean space with Euclidean norm, this corresponds to the Aronsson equation with Hamiltonian \[ H(u, \nabla u) = |\nabla u| + \beta u, \] equivalently viscosity solutions of $\Delta_{\infty}^{\beta} u = 0$. We show that $\beta$-AM arises as the continuum value of a biased tug-of-war game. Analogous to the unbiased case, we derive various properties of this extension. As an application, we further show that the linear blow-up property holds for biased infinity harmonic functions.

math.AP↗

arXiv2024

Non-Markovianity of $2K-B$ and a degeneration

We study the process of $2K-B$, where $B$ is a standard one-dimensional Brownian motion and $K$ is its concave majorant. In light of Pitman's $2M-B$ theorem, it was recently conjectured by Ouaki and Pitman \cite{OP} that $2K-B$ has the law of the BES(5) process. The two processes share properties such as Brownian scaling, time inversion and quadratic variation, and the same one point distribution and infinitesimal generator, among many other evidences; and it remains to prove that $2K-B$ is Markovian. However, we show that this conjecture is false. To better understand the similarity between these two processes, we study a degeneration of $2K-B$. We show it is a mixture of BES(3), and get other properties including multiple points distribution, infinitesimal generator, and path decomposition at future infimum. We also further investigate the Markovian structure and the filtrations of $2K-B, B$ and $K$.

math.PR↗

arXiv2020

Learning spatial hearing via innate mechanisms

The acoustic cues used by humans and other animals to localise sounds are subtle, and change during and after development. This means that we need to constantly relearn or recalibrate the auditory spatial map throughout our lifetimes. This is often thought of as a "supervised" learning process where a "teacher" (for example, a parent, or your visual system) tells you whether or not you guessed the location correctly, and you use this information to update your map. However, there is not always an obvious teacher (for example in babies or blind people). Using computational models, we showed that approximate feedback from a simple innate circuit, such as that can distinguish left from right (e.g. the auditory orienting response), is sufficient to learn an accurate full-range spatial auditory map. Moreover, using this mechanism in addition to supervised learning can more robustly maintain the adaptive neural representation. We find several possible neural mechanisms that could underlie this type of learning, and hypothesise that multiple mechanisms may be present and interact with each other. We conclude that when studying spatial hearing, we should not assume that the only source of learning is from the visual system or other supervisory signal. Further study of the proposed mechanisms could allow us to design better rehabilitation programmes to accelerate relearning/recalibration of spatial maps.

cs.NE↗

arXiv2019

Identify Equivalent Frames

A frame is an overcomplete set that can represent vectors(signals) faithfully and stably. Two frames are equivalent if signals can be essentially represented in the same way, which means two frames differ by a permutation, sign change or orthogonal transformation. Since these operations are combinatorial in nature, it is infeasible to check whether two frames are equivalent by exhaustive search. In this note, we present an algorithm that can check this equivalence in polynomial time. Theoretical guarantees are provided for special cases.

cs.IT↗

arXiv2019

Solution of the 15 puzzle problem

A generalized `$15$ puzzle' consists of an $n \times n$ numbered grid, with one missing number. A move in the game switches the position of the empty square with the position of one of its neighbors. We solve Diaconis' `15 puzzle problem' by proving that the asymptotic total variation mixing time of the board is at least order $ n^4 $ when the board is given periodic boundary conditions and when random moves are made. We demonstrate that for any $f(n) \to \infty$ with $n$, the number of fixed points after $n^4 f(n)$ moves converges to a Poisson distribution of parameter 1. The order of total variation mixing time for this convergence is $n^4$ without cut-off. We also prove an upper bound of order $n^{4 }\log n$ for the total variation mixing time.

math.PR↗