483 research outputs found

### A Distributed Tracking Algorithm for Reconstruction of Graph Signals

The rapid development of signal processing on graphs provides a new
perspective for processing large-scale data associated with irregular domains.
In many practical applications, it is necessary to handle massive data sets
through complex networks, in which most nodes have limited computing power.
Designing efficient distributed algorithms is critical for this task. This
paper focuses on the distributed reconstruction of a time-varying bandlimited
graph signal based on observations sampled at a subset of selected nodes. A
distributed least square reconstruction (DLSR) algorithm is proposed to recover
the unknown signal iteratively, by allowing neighboring nodes to communicate
with one another and make fast updates. DLSR uses a decay scheme to annihilate
the out-of-band energy occurring in the reconstruction process, which is
inevitably caused by the transmission delay in distributed systems. Proof of
convergence and error bounds for DLSR are provided in this paper, suggesting
that the algorithm is able to track time-varying graph signals and perfectly
reconstruct time-invariant signals. The DLSR algorithm is numerically
experimented with synthetic data and real-world sensor network data, which
verifies its ability in tracking slowly time-varying graph signals.Comment: 30 pages, 9 figures, 2 tables, journal pape

### Estimation of Markov Chain via Rank-Constrained Likelihood

This paper studies the estimation of low-rank Markov chains from empirical
trajectories. We propose a non-convex estimator based on rank-constrained
likelihood maximization. Statistical upper bounds are provided for the
Kullback-Leiber divergence and the $\ell_2$ risk between the estimator and the
true transition matrix. The estimator reveals a compressed state space of the
Markov chain. We also develop a novel DC (difference of convex function)
programming algorithm to tackle the rank-constrained non-smooth optimization
problem. Convergence results are established. Experiments show that the
proposed estimator achieves better empirical performance than other popular
approaches.Comment: Accepted at ICML 201

### Waring-Goldbach problem in short intervals

Let $k\geq2$ and $s$ be positive integers. Let $\theta\in(0,1)$ be a real
number. In this paper, we establish that if $s>k(k+1)$ and $\theta>0.55$, then
every sufficiently large natural number $n$, subjects to certain congruence
conditions, can be written as $n=p_1^k+\cdots+p_s^k,$ where $p_i(1\leq
i\leq s)$ are primes in the interval
$((\frac{n}{s})^{\frac{1}{k}}-n^{\frac{\theta}{k}},(\frac{n}{s})^{\frac{1}{k}}+n^{\frac{\theta}{k}}]$.
The second result of this paper is to show that if $s>\frac{k(k+1)}{2}$ and
$\theta>0.55$, then almost all integers $n$, subject to certain congruence
conditions, have above representation.Comment: 18 page

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