ABSTRACT If $X$ is a random variable with $EX^2 = \sigma^2$, then by Chebyshev's inequality, \begin{equation*}\tag{1.1}P\{|X| \geqq \epsilon\} \leqq \sigma^2/\epsilon^2.\end{equation*} If in addition $EX = 0$, one obtains a corresponding one-sided inequality \begin{equation*}\tag{1.2}\quad P\{X \geqq \epsilon\} \leqq \sigma^2/ (\epsilon^2 + \sigma^2)\end{equation*} (see, e.g., [8] p. 198). In each case a distribution for $X$ is known that results in equality, so that the bounds are sharp. By a change of variable we can take $\epsilon = 1$. There are many possible multivariate extensions of (1.1) and (1.2). Those providing bounds for $P\{\max_{1 \leqq j \leqq k} |X_j| \geqq 1\}$ and $P\{|\max_{1 \leqq j \leqq k} X_j \geqq 1\}$ have been investigated in [3, 5, 9] and [4], respectively. We consider here various inequalities involving (i) the minimum component or (ii) the product of the components of a random vector. Derivations and proofs of sharpness for these two classes of extensions show remarkable similarities. Some of each type occur as special cases of a general theorem in Section 3. Bounds are given under various assumptions concerning variances, covariances and independence. Notation. We denote the vector $(1, \cdots, 1)$ by $e$ and $(0, \cdots, 0)$ by 0; the dimensionality will be clear from the context. If $x = (x_1, \cdots, x_k)$ and $y = (y_1, \cdots, y_k)$, we write $x \geqq y(x > y)$ to mean $x_j \geqq y_j(x_j > y_j), j = 1, 2, \cdots, k$. If $\Sigma = (\sigma_{ij}): k \times k$ is a moment matrix, for convenience we write $\sigma_{jj} = \sigma^2_j, j = 1, \cdots, k$. Unless otherwise stated, we assume that $\Sigma$ is positive definite.
[Show abstract][Hide abstract] ABSTRACT:
This paper proposes a new approach for discriminative clustering. The intuition is, for a good clustering, one should be able to learn a classifier from the clustering labels with high generalization accuracy. Thus we define a novel metric to evaluate the quality of a clustering labeling, named Minimum Separation Probability (MSP), which is a lower bound of the generalization accuracy of a classifier learnt from the clustering labeling. We take MSP as the objective to maximize and propose our approach Maximin Separation Probability Clustering (MSPC), which has several attractive properties, such as invariance to anisotropic feature scaling and intuitive probabilistic explanation for clustering quality. We present three efficient optimization strategies for MSPC, and analyze their interesting connections to existing clustering approaches, such as maximum margin clustering (MMC) and discriminative k-means. Empirical results on real world data sets verify that MSP is a robust and effective clustering quality measure. It is also shown that the proposed algorithms compare favorably to state-of-the-art clustering algorithms in both accuracy and efficiency.
AAAI Conference on Artificial Intelligence; 01/2015
[Show abstract][Hide abstract] ABSTRACT:
In this paper, we propose a maximum margin classifier that deals with
uncertainty in data input. Specifically, we reformulate the SVM framework such
that each input training entity is not solely a feature vector representation,
but a multi-dimensional Gaussian distribution with given probability density,
i.e., with a given mean and covariance matrix. The latter expresses the
uncertainty. We arrive at a convex optimization problem, which is solved in the
primal form using a gradient descent approach. The resulting classifier, which
we name SVM with Gaussian Sample Uncertainty (SVM-GSU), is tested on synthetic
data, as well as on the problem of event detection in video using the
large-scale TRECVID MED 2014 dataset, and the problem of image classification
using the MNIST dataset of handwritten digits. Experimental results verify the
effectiveness of the proposed classifier.
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