460 research outputs found
Hilbert series and Hilbert depth of squarefree Veronese ideals
In this paper, we obtain explicit formulas for the Hilbert series and Hilbert
depth of squarefree Veronese ideals in a standard graded polynomial ring.Comment: 7 pages, a gap in the previous version is fixe
Declutter and Resample: Towards parameter free denoising
In many data analysis applications the following scenario is commonplace: we
are given a point set that is supposed to sample a hidden ground truth in a
metric space, but it got corrupted with noise so that some of the data points
lie far away from creating outliers also termed as {\em ambient noise}. One
of the main goals of denoising algorithms is to eliminate such noise so that
the curated data lie within a bounded Hausdorff distance of . Popular
denoising approaches such as deconvolution and thresholding often require the
user to set several parameters and/or to choose an appropriate noise model
while guaranteeing only asymptotic convergence. Our goal is to lighten this
burden as much as possible while ensuring theoretical guarantees in all cases.
Specifically, first, we propose a simple denoising algorithm that requires
only a single parameter but provides a theoretical guarantee on the quality of
the output on general input points. We argue that this single parameter cannot
be avoided. We next present a simple algorithm that avoids even this parameter
by paying for it with a slight strengthening of the sampling condition on the
input points which is not unrealistic. We also provide some preliminary
empirical evidence that our algorithms are effective in practice
Scalable Multi-Parameter Outlier Detection Technology
The real-time detection of anomalous phenomena on streaming data has become increasingly important for applications ranging from fraud detection, financial analysis to traffic management. In these streaming applications, often a large number of similar continuous outlier detection queries are executed concurrently. In the light of the high algorithmic complexity of detecting and maintaining outlier patterns for different parameter settings independently, we propose a shared execution methodology called SOP that handles a large batch of requests with diverse pattern configurations. First, our systematic analysis reveals opportunities for maximum resource sharing by leveraging commonalities among outlier detection queries. For that, we introduce a sharing strategy that integrates all computation results into one compact data structure. It leverages temporal relationships among stream data points to prioritize the probing process. Second, this work is the first to consider predicate constraints in the outlier detection context. By distinguishing between target and scope constraints, customized fragment sharing and block selection strategies can be effectively applied to maximize the efficiency of system resource utilization. Our experimental studies utilizing real stream data demonstrate that our approach performs 3 orders of magnitude faster than the start-of-the-art and scales to 1000s of queries
Spectral Efficiency and Energy Efficiency Tradeoff in Massive MIMO Downlink Transmission with Statistical CSIT
As a key technology for future wireless networks, massive multiple-input
multiple-output (MIMO) can significantly improve the energy efficiency (EE) and
spectral efficiency (SE), and the performance is highly dependant on the degree
of the available channel state information (CSI). While most existing works on
massive MIMO focused on the case where the instantaneous CSI at the transmitter
(CSIT) is available, it is usually not an easy task to obtain precise
instantaneous CSIT. In this paper, we investigate EE-SE tradeoff in single-cell
massive MIMO downlink transmission with statistical CSIT. To this end, we aim
to optimize the system resource efficiency (RE), which is capable of striking
an EE-SE balance. We first figure out a closed-form solution for the
eigenvectors of the optimal transmit covariance matrices of different user
terminals, which indicates that beam domain is in favor of performing RE
optimal transmission in massive MIMO downlink. Based on this insight, the RE
optimization precoding design is reduced to a real-valued power allocation
problem. Exploiting the techniques of sequential optimization and random matrix
theory, we further propose a low-complexity suboptimal two-layer
water-filling-structured power allocation algorithm. Numerical results
illustrate the effectiveness and near-optimal performance of the proposed
statistical CSI aided RE optimization approach.Comment: Typos corrected. 14 pages, 7 figures. Accepted for publication on
IEEE Transactions on Signal Processing. arXiv admin note: text overlap with
arXiv:2002.0488
Coincidences between intervals in two partial orders on complex reflection groups
In a finite real reflection group, the reflection length of each element is
equal to the codimension of its fixed space, and the two coincident functions
determine a partial order structure called the absolute order. In complex
reflection groups, the reflection length is no longer always equal to the
codimension of fixed space, and the two functions give rise to two different
partial orders on the group. We characterize the elements in the
combinatorial family of complex reflection groups for which the
intervals below in these two posets coincide.Comment: 15 page
Graph Reconstruction by Discrete Morse Theory
Recovering hidden graph-like structures from potentially noisy data is a fundamental task in modern data analysis. Recently, a persistence-guided discrete Morse-based framework to extract a geometric graph from low-dimensional data has become popular. However, to date, there is very limited theoretical understanding of this framework in terms of graph reconstruction. This paper makes a first step towards closing this gap. Specifically, first, leveraging existing theoretical understanding of persistence-guided discrete Morse cancellation, we provide a simplified version of the existing discrete Morse-based graph reconstruction algorithm. We then introduce a simple and natural noise model and show that the aforementioned framework can correctly reconstruct a graph under this noise model, in the sense that it has the same loop structure as the hidden ground-truth graph, and is also geometrically close. We also provide some experimental results for our simplified graph-reconstruction algorithm
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