A note on correlation and local dimensions
Under very mild assumptions, we give formulas for the correlation and local dimensions of measures on the limit set of a Moran construction by means of the data used to construct the set.
math.CA↗
arXiv subjects
Min Wu
Publications and source records attributed to Min Wu.
Sampling Requirements for Stable Autoregressive Estimation
We consider the problem of estimating the parameters of a linear univariate autoregressive model with sub-Gaussian innovations from a limited sequence of consecutive observations. Assuming that the parameters are compressible, we analyze the performance of the $\ell_1$-regularized least squares as well as a greedy estimator of the parameters and characterize the sampling trade-offs required for stable recovery in the non-asymptotic regime. In particular, we show that for a fixed sparsity level, stable recovery of AR parameters is possible when the number of samples scale sub-linearly with the AR order. Our results improve over existing sampling complexity requirements in AR estimation using the LASSO, when the sparsity level scales faster than the square root of the model order. We further derive sufficient conditions on the sparsity level that guarantee the minimax optimality of the $\ell_1$-regularized least squares estimate. Applying these techniques to simulated data as well as real-world datasets from crude oil prices and traffic speed data confirm our predicted theoretical performance gains in terms of estimation accuracy and model selection.
cs.IT↗
The topological property of the irregular sets on the lengths of basic intervals in beta-expansions
Let $β> 1$ be a real number and $(ε_1(x, β), ε_2(x, β), \ldots)$ be the $β$-expansion of a point $x \in (0, 1]$. For all $x \in (0,1]$, let $A(D(x))$ be the set of accumulation points of $\frac{-\log_β|I_n(x)|}{n}$ as $n \rightarrow \infty$, where $|I_n(x)|$ is the length of the basic interval of order $n$ containing $x \in (0, 1]$. In this paper, we prove that $A(D(x))$ is always a closed interval for any $x \in (0,1]$. Furthermore, if $λ(β)>0$, the extremely irregular set containing points $x \in [0, 1]$ whose upper limit of $\frac{-\log_β|I_n(x)|}{n}$ equals to $1+ł(β)$ is residual, where $1+ł(β)$ is a constant depending on $β$. As a consequence, the irregular set with $x\in [0, 1]$ whose limit of $\frac{-\log_β|I_n(x)|}{n}$ does not exist is residual for every $λ(β)>0$.
math.DS↗
The dimension of irregular set in parameter space
For any real number $β>1$. The $n$th cylinder of $β$ in the parameter space $\{β\in \mathbb{R}: β>1\}$ is a set of real numbers in $(1,\infty)$ having the same first $n$ digits in their $β$-expansion of $1$, denote by $I^P_n(β)$. We study the quantities which describe the growth of the length of $I^P_n(β)$. The Huasdorff dimension of the set of given growth rate of the length of $I^P_n(β)$ will be determined in this paper.
math.DS↗
Efficient Estimation of Compressible State-Space Models with Application to Calcium Signal Deconvolution
In this paper, we consider linear state-space models with compressible innovations and convergent transition matrices in order to model spatiotemporally sparse transient events. We perform parameter and state estimation using a dynamic compressed sensing framework and develop an efficient solution consisting of two nested Expectation-Maximization (EM) algorithms. Under suitable sparsity assumptions on the innovations, we prove recovery guarantees and derive confidence bounds for the state estimates. We provide simulation studies as well as application to spike deconvolution from calcium imaging data which verify our theoretical results and show significant improvement over existing algorithms.
stat.ML↗
The denominators of convergents for continued fractions
For any real number $x \in [0,1)$, we denote by $q_n(x)$ the denominator of the $n$-th convergent of the continued fraction expansion of $x$ $(n \in \mathbb{N})$. It is well-known that the Lebesgue measure of the set of points $x \in [0,1)$ for which $\log q_n(x)/n$ deviates away from $π^2/(12\log2)$ decays to zero as $n$ tends to infinity. In this paper, we study the rate of this decay by giving an upper bound and a lower bound. What is interesting is that the upper bound is closely related to the Hausdorff dimensions of the level sets for $\log q_n(x)/n$. As a consequence, we obtain a large deviation type result for $\log q_n(x)/n$, which indicates that the rate of this decay is exponential.
math.NT↗
Impact Analysis of Baseband Quantizer on Coding Efficiency for HDR Video
Digitally acquired high dynamic range (HDR) video baseband signal can take 10 to 12 bits per color channel. It is economically important to be able to reuse the legacy 8 or 10-bit video codecs to efficiently compress the HDR video. Linear or nonlinear mapping on the intensity can be applied to the baseband signal to reduce the dynamic range before the signal is sent to the codec, and we refer to this range reduction step as a baseband quantization. We show analytically and verify using test sequences that the use of the baseband quantizer lowers the coding efficiency. Experiments show that as the baseband quantizer is strengthened by 1.6 bits, the drop of PSNR at a high bitrate is up to 1.60dB. Our result suggests that in order to achieve high coding efficiency, information reduction of videos in terms of quantization error should be introduced in the video codec instead of on the baseband signal.
cs.MM↗
A note on Rényi's "record" problem and Engel's series
In 1973, Williams introduced two interesting discrete Markov processes, namely $C$-processes and $A$-processes, which are related to record times in statistics and Engel's series in number theory respectively. Moreover, he showed that these two processes share the same classical limit theorems, such as the law of large numbers, central limit theorem and law of the iterated logarithm. In this paper, we consider the large deviations for these two Markov processes, which indicate that there is a difference between $C$-processes and $A$-processes in the context of large deviations.
math.ST↗
Approximation orders of real numbers by $β$-expansions
We prove that almost all real numbers (with respect to Lebesgue measure) are approximated by the convergents of their $β$-expansions with the exponential order $β^{-n}$. Moreover, the Hausdorff dimensions of sets of the real numbers which are approximated by all other orders, are determined. These results are also applied to investigate the orbits of real numbers under $β$-transformation, the shrinking target type problem, the Diophantine approximation and the run-length function of $β$-expansions.
math.NT↗
Learning Optimal Social Dependency for Recommendation
Social recommender systems exploit users' social relationships to improve the recommendation accuracy. Intuitively, a user tends to trust different subsets of her social friends, regarding with different scenarios. Therefore, the main challenge of social recommendation is to exploit the optimal social dependency between users for a specific recommendation task. In this paper, we propose a novel recommendation method, named probabilistic relational matrix factorization (PRMF), which aims to learn the optimal social dependency between users to improve the recommendation accuracy, with or without users' social relationships. Specifically, in PRMF, the latent features of users are assumed to follow a matrix variate normal (MVN) distribution. The positive and negative dependency between users are modeled by the row precision matrix of the MVN distribution. Moreover, we have also proposed an efficient alternating algorithm to solve the optimization problem of PRMF. The experimental results on real datasets demonstrate that the proposed PRMF method outperforms state-of-the-art social recommendation approaches, in terms of root mean square error (RMSE) and mean absolute error (MAE).
cs.IR↗
Beta-expansion and continued fraction expansion of real numbers
Let $β> 1$ be a real number and $x \in [0,1)$ be an irrational number. We denote by $k_n(x)$ the exact number of partial quotients in the continued fraction expansion of $x$ given by the first $n$ digits in the $β$-expansion of $x$ ($n \in \mathbb{N}$). It is known that $k_n(x)/n$ converges to $(6\log2\logβ)/π^2$ almost everywhere in the sense of Lebesgue measure. In this paper, we improve this result by proving that the Lebesgue measure of the set of $x \in [0,1)$ for which $k_n(x)/n$ deviates away from $(6\log2\logβ)/π^2$ decays to 0 exponentially as $n$ tends to $\infty$, which generalizes the result of Faivre \cite{lesFai97} from $β= 10$ to any $β>1$. Moreover, we also discuss which of the $β$-expansion and continued fraction expansion yields the better approximations of real numbers.
math.NT↗
On exceptional sets in Erdős-Rényi limit theorem revisited
For $x\in [0,1],$ the run-length function $r_n(x)$ is defined as the length of the longest run of $1$'s amongst the first $n$ dyadic digits in the dyadic expansion of $x.$ Erdős and Rényi proved that $\lim\limits_{n\to\infty}\frac{r_n(x)}{\log_2n}=1$ for Lebesgue almost all $x\in[0,1]$. Let $H$ denote the set of monotonically increasing functions $φ:\mathbb{N}\to (0,+\infty)$ with $\lim\limits_{n\to\infty}φ(n)=+\infty$. For any $φ\in H$, we prove that the set \[ E_{\max}^φ=\left\{x\in [0,1]:\liminf\limits_{n\to\infty}\frac{r_n(x)}{φ(n)}=0, \limsup\limits_{n\to\infty}\frac{r_n(x)}{φ(n)}=+\infty\right\} \] either has Hausdorff dimension one and is residual in $[0,1]$ or empty. The result solves a conjecture posed in \cite{LW5} affirmatively.
math.PR↗
Limit theorems related to beta-expansion and continued fraction expansion
Let $β> 1$ be a real number and $x \in [0,1)$ be an irrational number. Denote by $k_n(x)$ the exact number of partial quotients in the continued fraction expansion of $x$ given by the first $n$ digits in the $β$-expansion of $x$ ($n \in \mathbb{N}$). In this paper, we show a central limit theorem and a law of the iterated logarithm for the random variables sequence $\{k_n, n \geq 1\}$, which generalize the results of Faivre and Wu respectively from $β=10$ to any $β>1$.
math.NT↗
Random Continued fractions: Lévy constant and Chernoff-type estimate
Given a stochastic process $\{A_n, n \geq 1\}$ taking values in natural numbers, the random continued fractions is defined as $[A_1, A_2, \cdots, A_n, \cdots]$ analogue to the continued fraction expansion of real numbers. Assume that $\{A_n, n \geq 1\}$ is ergodic and the expectation $E(\log A_1) < \infty$, we give a Lévy-type metric theorem which covers that of real case presented by Lévy in 1929. Moreover, a corresponding Chernoff-type estimate is obtained under the conditions $\{A_n, n \geq 1\}$ is $ψ$-mixing and for each $0< t< 1$, $E(A_1^t) < \infty$.
math.NT↗
Multi-label ensemble based on variable pairwise constraint projection
Multi-label classification has attracted an increasing amount of attention in recent years. To this end, many algorithms have been developed to classify multi-label data in an effective manner. However, they usually do not consider the pairwise relations indicated by sample labels, which actually play important roles in multi-label classification. Inspired by this, we naturally extend the traditional pairwise constraints to the multi-label scenario via a flexible thresholding scheme. Moreover, to improve the generalization ability of the classifier, we adopt a boosting-like strategy to construct a multi-label ensemble from a group of base classifiers. To achieve these goals, this paper presents a novel multi-label classification framework named Variable Pairwise Constraint projection for Multi-label Ensemble (VPCME). Specifically, we take advantage of the variable pairwise constraint projection to learn a lower-dimensional data representation, which preserves the correlations between samples and labels. Thereafter, the base classifiers are trained in the new data space. For the boosting-like strategy, we employ both the variable pairwise constraints and the bootstrap steps to diversify the base classifiers. Empirical studies have shown the superiority of the proposed method in comparison with other approaches.
cs.LG↗
In-situ EXAFS study on the thermal decomposition of TiH2
Thermal decomposition behaviors of TiH2 powder under a flowing helium atmosphere and in a low vacuum condition have been studied by using in-situ EXAFS technique. By an EXAFS analysis containing the multiple scattering paths including H atoms, the changes of hydrogen stoichiometric ratio and the phase transformation sequence are obtained. The results demonstrate that the initial decomposition temperature is dependent on experimental conditions, which occurs, respectively, at about 300 and 400 degree in a low vacuum condition and under a flowing helium atmosphere. During the decomposition process of TiH2 in a low vacuum condition, the sample experiences a phase change process: δ(TiH2) - δ(TiHx) - δ(TiHx)+β(TiHx) - δ(TiHx)+β(TiHx)+α(Ti) - β(TiHx)+α(Ti) - α(Ti)+β(Ti). This study offers a way to detect the structural information of hydrogen. A detailed discussion about the decomposition process of TiH2 is given in this paper.
cond-mat.mtrl-sci↗
Sparse Norm Filtering
Optimization-based filtering smoothes an image by minimizing a fidelity function and simultaneously preserves edges by exploiting a sparse norm penalty over gradients. It has obtained promising performance in practical problems, such as detail manipulation, HDR compression and deblurring, and thus has received increasing attentions in fields of graphics, computer vision and image processing. This paper derives a new type of image filter called sparse norm filter (SNF) from optimization-based filtering. SNF has a very simple form, introduces a general class of filtering techniques, and explains several classic filters as special implementations of SNF, e.g. the averaging filter and the median filter. It has advantages of being halo free, easy to implement, and low time and memory costs (comparable to those of the bilateral filter). Thus, it is more generic than a smoothing operator and can better adapt to different tasks. We validate the proposed SNF by a wide variety of applications including edge-preserving smoothing, outlier tolerant filtering, detail manipulation, HDR compression, non-blind deconvolution, image segmentation, and colorization.
cs.GR↗
Domain-of-Attraction Estimation for Uncertain Non-polynomial Systems
In this paper, we consider the problem of computing estimates of the domain-of-attraction for non-polynomial systems. A polynomial approximation technique, based on multivariate polynomial interpolation and error analysis for remaining functions, is applied to compute an uncertain polynomial system, whose set of trajectories contains that of the original non-polynomial system. Experiments on the benchmark non-polynomial systems show that our approach gives better estimates of the domain-of-attraction.
cs.SC↗