Using Theorem Proving to Verify Expectation and Variance for Discrete Random Variables

Statistical quantities, such as expectation (mean) and variance, play a vital role in the present age probabilistic analysis. In this paper, we present some formalization of expectation theory that can be used to verify the expectation and variance characteristics of discrete random variables within the HOL theorem prover. The motivation behind this is the ability to perform error free probabilistic analysis, which in turn can be very useful for the performance and reliability analysis of systems used in safety-critical domains, such as space travel, medicine and military. We first present a formal definition of expectation of a function of a discrete random variable. Building upon this definition, we formalize the mathematical concept of variance and verify some classical properties of expectation and variance in HOL. We then utilize these formal definitions to verify the expectation and variance characteristics of the Geometric random variable. In order to demonstrate the practical effectiveness of the formalization presented in this paper, we also present the probabilistic analysis of the Coupon Collector’s problem in HOL.     

Locally Minimizing Embedding and Globally Maximizing Variance: Unsupervised Linear Difference Projection for Dimensionality Reduction

Recently, many dimensionality reduction algorithms, including local methods and global methods, have been presented. The representative local linear methods are locally linear embedding (LLE) and linear preserving projections (LPP), which seek to find an embedding space that preserves local information to explore the intrinsic characteristics of high dimensional data. However, both of them still fail to nicely deal with the sparsely sampled or noise contaminated datasets, where the local neighborhood structure is critically distorted. On the contrary, principal component analysis (PCA), the most frequently used global method, preserves the total variance by maximizing the trace of feature variance matrix. But PCA cannot preserve local information due to pursuing maximal variance. In order to integrate the locality and globality together and avoid the drawback in LLE and PCA, in this paper, inspired by the dimensionality reduction methods of LLE and PCA, we propose a new dimensionality reduction method for face recognition, namely, unsupervised linear difference projection (ULDP). This approach can be regarded as the integration of a local approach (LLE) and a global approach (PCA), so that it has better performance and robustness in applications. Experimental results on the ORL, YALE and AR face databases show the effectiveness of the proposed method on face recognition.     

低姿态回归职场

米卢说“态度决定一切”。这不仅是足球的态度,也是人生的态度。2004年的欧洲杯给我上了人生中重要的一课,让我低姿态回归了我的人生职场。我来自南方一个偏远的农村,家乡的交通很闭塞。乡亲们主要靠种地为生,过着面朝黄土背朝天的生活。2000年,我考入了一所重点大学的计算机专业,一下子成了村里飞出来的金凤凰。当时计算机还算比较热门的专业,华为、中兴等国内知名企业,微软、通用等跨国公司还纷纷来到学校高薪招聘。那时,我在心里暗暗给自己订了一个目标,一定要进入国内最优秀的IT企业,摆脱贫穷的日子,成为业内的精英。这是我给自己绘制…     

低姿态回归职场

米卢说“态度决定一切”。这不仅是足球的态度，也是人生的态度。2004年的欧洲杯给我上了人生中重要的一课，让我低姿态回归了我的人生职场。     

Minimizing FPGA interconnect delays

Optimizing FPGA routing architectures for speed performance also involves improving the CAD tools for mapping circuits. We provide a detailed example of how to design FPGA architectures by examining several important issues associated with interconnect resources for FPGAs that use SRAM programming technology. Our experiments examine two important metrics: the speed performance of implemented circuits and the effective use of available interconnect resources. The goal is to improve upon FPGA speed performance by decreasing delays associated with the interconnect. Our results are most directly applicable to FPGA architectures similar in style to the Xilinx XC4000 series. However, some significant results are of a more general nature and perhaps applicable to other styles of FPGAs as well. In addition to routing architectures, we address the CAD tools that allocate these routing resources to implement circuits     

模拟的概念、计算及在自动机最小化上的应用

对Bachi自动机进行优化是提高基于自动机的模型检测效率的重要手段。本文对直接模拟关系，延迟模拟关系和公平模拟关系的概念，算法进行了比较，并探讨了基于这些模拟关系的自动机优化方法。最后对未来的研究方向作了简要的介绍。     

Minimizing hierarchical routing error

In hierarchical routing schemes, nodes are grouped into clusters at multiple levels, and a given node sees only a summarized view of the entire network. Hierarchical routing introduces error, which is the difference between the hierarchical path length and the optimal path length using flat routing. Since in practice the routing table size at each node is limited, we formulate the constrained optimization problems of finding a hierarchy structure that minimizes either the worst case or average case routing error. We prove results characterizing solutions of these problems, and present dynamic programming solution algorithms and computational results.     

Love Plus

在游戏中第天被分为4个时间段,和Konaml的经典恋爱游戏“《心跳回忆》系列”一样,本作同样需要玩家自己安排课程来增加各项能力。     

Minimizing Energies with Hierarchical Costs

Computer vision is full of problems elegantly expressed in terms of energy minimization. We characterize a class of energies with hierarchical costs and propose a novel hierarchical fusion algorithm. Hierarchical costs are natural for modeling an array of difficult problems. For example, in semantic segmentation one could rule out unlikely object combinations via hierarchical context. In geometric model estimation, one could penalize the number of unique model families in a solution, not just the number of models??a kind of hierarchical MDL criterion. Hierarchical fusion uses the well-known ??-expansion algorithm as a subroutine, and offers a much better approximation bound in important cases.     

Determining Knots by Minimizing Energy

A new method for determining knots to construct polynomial curves is presented. At each data point, a quadric curve which passes three consecutive points is constructed. The knots for constructing the quadric curve are determined by minimizing the internal strain energy, which can be regarded as a function of the angle. The function of the angle is expanded as a Taylor series with two terms, then the two knot intervals between the three consecutive points are defined by linear expression. Between the two consecutive points, there are two knot intervals, and the combination of the two knot intervals is used to define the final knot interval. A comparison of the new method with several existing methods is included.     

Minimizing representations over number fields

Finding minimal fields of definition for representations is one of the most important unsolved problems of computational representation theory. While good methods exist for representations over finite fields, it is still an open question in the case of number fields. In this paper we give a practical method for finding minimal defining subfields for absolutely irreducible representations. We illustrate the new algorithm by determining a minimal field for each absolutely irreducible representation of Sz(8).     

Variance constrained self-tuning control

A self-tuning regulator for a variance constrained optimal control problem is given. The criterion for control is to minimize the stationary variance of the output. In the cases when the regulator which gives minimum variance requires too large control signals an inequality constraint on the input variance is introduced. In practice it is easier to select a constraint on the variance of the input than to choose the relative weights in a quadratic loss function. The self-tuning regulator applies the Robbins-Monro scheme to adjust the Lagrange multiplier of the variance constrained control problem. The behaviour of the algorithm is illustrated by a simulated example. The asymptotic behaviour of the regulator is studied using a set of associated ordinary differential equations.     

Variance enhanced K-medoid clustering

This paper proposes new variance enhanced clustering methods to improve the popular K-medoid algorithm by adapting variance information in data clustering. Since measuring similarity between data objects is simpler than mapping data objects to data points in feature space, these pairwise similarity based clustering algorithms can greatly reduce the difficulty in developing clustering based pattern recognition applications. A web-based image clustering system has been developed to demonstrate and show the clustering power and significance of the proposed methods. Synthetic numerical data and real-world image collection are applied to evaluate the performance of the proposed methods on the prototype system. As shown as the web-demonstration, the proposed method, variance enhanced K-medoid model, groups similar images in clusters with various variances according to the distribution of image similarity values.     

Variance Soft Shadow Mapping

We present variance soft shadow mapping (VSSM) for rendering plausible soft shadow in real‐time. VSSM is based on the theoretical framework of percentage‐closer soft shadows (PCSS) and exploits recent advances in variance shadow mapping (VSM). Our new formulation allows for the efficient computation of (average) blocker distances, a common bottleneck in PCSS‐based methods. Furthermore, we avoid incorrectly lit pixels commonly encountered in VSM‐based methods by appropriately subdividing the filter kernel. We demonstrate that VSSM renders high‐quality soft shadows efficiently (usually over 100 fps) for complex scene settings. Its speed is at least one order of magnitude faster than PCSS for large penumbra.