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48532 results in Computer Science

Page 551 of 2427

  • 13 - The Normal Distribution

  • from Part V - Advanced Topics
  • Jianxin Wu, Nanjing University, China
  • Book:
    Essentials of Pattern Recognition
    Published online:
    08 December 2020
    Print publication:
    19 November 2020, pp 293-315

  • Summary

    The normal distribution is the most widely used continuous distribution, but many of its relevant properties are a little bit advanced for an undergraduate course. Hence, Part IV introduces some of these advanced topics. This chapter devotes itself to properties of normal distributions: single- and multivariate normal distributions, moment and canonical parameterizations, sum and product, geometry and the Mahalanobis distance, and conditional distributions. We also show that with these properties, some algorithms will become much easier to understand. We use parameter estimation and the Kalman filter as two such examples.

  • 15 - Convolutional Neural Networks

  • from Part V - Advanced Topics
  • Jianxin Wu, Nanjing University, China
  • Book:
    Essentials of Pattern Recognition
    Published online:
    08 December 2020
    Print publication:
    19 November 2020, pp 333-364

  • Summary

    We cannot miss deep learning in a modern pattern recognition textbook, and we introduce CNN (convolutional neural networks) in this chapter. Although the mathematical derivation of CNN, especially the back-propagation process and gradient computation, is complex, we use a lot of useful tools to help readers understand what exactlyis going on in a CNN. Hence, this chapter focuses on accessibility rather than completeness. In its exercise problems, we introduce more relevant topics and methods.

  • 6 - Fisher’s Linear Discriminant

  • from Part II - Domain-Independent Feature Extraction
  • Jianxin Wu, Nanjing University, China
  • Book:
    Essentials of Pattern Recognition
    Published online:
    08 December 2020
    Print publication:
    19 November 2020, pp 123-140

  • Summary

    Unlike PCA, which is unsupervised, FLD uses labels associated with data points, and no doubt it may get better linear features and accuracy than PCA. We start by illustrating this motivation, and practice the problem-solving framework by gradually developing the correct mathematical formulation behind the relatively simple idea behind Fisher's linear discriminant (FLD). We discuss various practical issues: the solution for the binary case, the scenario where this solution breaks down, and how to generalize from tasks with only two categories to many categories.

  • 8 - Probabilistic Methods

  • from Part III - Classifiers and Tools
  • Jianxin Wu, Nanjing University, China
  • Book:
    Essentials of Pattern Recognition
    Published online:
    08 December 2020
    Print publication:
    19 November 2020, pp 173-195

  • Summary

    This chapter is a succinct introduction to basic probabilistic methods for pattern recognition and machine learning. One focus is to clearly present the exact meanings of different terms, including the taxonomy of different probabilistic methods. We present a basic introduction to maximum likelihood and maximum a posteriori estimation, and a very brief example to showcase the concept of Bayesian estimation. For the nonparametric world, we start from the drawbacks of parametric methods, gradually analyzing the properties preferred for a nonparametric one, and finally reach the kernel density estimation, a typical nonparametric method.

  • 1 - Introduction

  • from Part I - Introduction and Overview
  • Jianxin Wu, Nanjing University, China
  • Book:
    Essentials of Pattern Recognition
    Published online:
    08 December 2020
    Print publication:
    19 November 2020, pp 3-14

  • Summary

    This chapter is an overall introduction to the definition of pattern recognition, its relationship with machine learning and other relevant subject areas, and the main components and development process inside a pattern recognition system. This introduction is started by considering an autonomous driving example.

  • Generalizations of the Ruzsa–Szemerédi and rainbow Turán problems for cliques

  • Part of
  • W. T. Gowers, Barnabás Janzer
  • Journal:
    Combinatorics, Probability and Computing / Volume 30 / Issue 4 / July 2021
    Published online by Cambridge University Press:
    19 November 2020, pp. 591-608
    • Article
      • You have access
      • Open access
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  • Considering a natural generalization of the Ruzsa–Szemerédi problem, we prove that for any fixed positive integers r, s with r < s, there are graphs on n vertices containing $n^{r}e^{-O\left(\sqrt{\log{n}}\right)}=n^{r-o(1)}$ copies of Ks such that any Kr is contained in at most one Ks. We also give bounds for the generalized rainbow Turán problem ex (n, H, rainbow - F) when F is complete. In particular, we answer a question of Gerbner, Mészáros, Methuku and Palmer, showing that there are properly edge-coloured graphs on n vertices with $n^{r-1-o(1)}$ copies of Kr such that no Kr is rainbow.

Page 551 of 2427