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
Data provided are for informational purposes only. Although carefully collected, accuracy cannot be guaranteed.
The impact factor represents a rough estimation of the journal's impact factor and does not reflect the actual
current impact factor.
Publisher conditions are provided by RoMEO. Differing provisions from the publisher's actual policy or licence
agreement may be applicable.
Page 1
Institute of Mathematical Statistics is collaborating with JSTOR to digitize, preserve, and extend access to The Annals of Mathematical Statistics. www.jstor.org ®