Robust loss functions are essential for training accurate deep neural networks (DNNs) in the presence of noisy (incorrect) labels. It has been shown that the commonly used Cross Entropy (CE) loss is not robust to noisy labels. Whilst new loss functions have been designed, they are only partially robust. In this paper, we theoretically show by applying a simple normalization that: \emph{any loss can be made robust to noisy labels}. However, in practice, simply being robust is not sufficient for a loss function to train accurate DNNs. By investigating several robust loss functions, we find that they suffer from a problem of \emph{underfitting}. To address this, we propose a framework to build robust loss functions called \emph{Active Passive Loss} (APL). APL combines two robust loss functions that mutually boost each other. Experiments on benchmark datasets demonstrate that the family of new loss functions created by our APL framework can consistently outperform state-of-the-art methods by large margins, especially under large noise rates such as 60% or 80% incorrect labels.

Correlation measures are a key element of statistics and machine learning, and essential for a wide range of data analysis tasks. Most existing correlation measures are for pairwise relationships, but real-world data can also exhibit complex multivariate correlations, involving three or more variables. We argue that multivariate correlation measures should be comparable, interpretable, scalable and unbiased. However, no existing measures satisfy all these requirements. In this paper, we propose an unbiased multivariate correlation measure, called UMC, which satisfies all the above criteria. UMC is a cumulative entropy based non-parametric multivariate correlation measure, which can capture both linear and non-linear correlations for groups of three or more variables. It employs a correction for chance using a statistical model of independence to address the issue of bias. UMC has high interpretability and we empirically show it outperforms state-of-the-art multivariate correlation measures in terms of statistical power, as well as for use in both subspace clustering and outlier detection tasks.

Previously, eight popular information-theoretic-based cluster validity indices have been generalized and tested for probabilistic partitions built by the expectation-maximization (EM) algorithm for the Gaussian mixture model. However, the analysis was limited to probabilistic clusters, and there were limited explanations for differences in the performance of the indices. In this paper, we extend the tests to partitions found by fuzzy c-means (FCM) and provide further explanations and insights about the performance of these indices. Of the eight generalized indices, we advocate a normalized version of the soft mutual information cluster validity index (NMI sM) as the best overall choice, as it outperforms the other seven indices for both FCM and EM according to our tests on synthetic and real data. The superiority of NMIsM is most pronounced for datasets with overlapped and/or varying-sized clusters. Finally, we provide a theoretical analysis, which helps explain the superior performance of NMIsM compared with the other three normalizations of soft mutual information.

Measuring the amount of dependency among multiple variables is an important task in pattern recognition. In the last few years, many new dependency measures have been developed for the exploration of functional relationships. In this paper, we develop a dependency measure between variables based on an extreme-value theoretic treatment of intrinsic dimensionality. Our measure identifies variables with low intrinsic dimension - that is, those that support embeddings of the data within low-dimensional manifolds. To build a dependency measure on strong foundations, we theoretically prove a connection between information theory and intrinsic dimensionality theory. This allows us also to propose novel estimators of intrinsic dimensionality. Finally, we show that our dependency measure enables to find patterns that cannot be found by other state-of-the-art measures on real and synthetic data.

Estimating the strength of dependency between two variables is fundamental for exploratory analysis and many other applications in data mining. For example: non-linear dependencies between two continuous variables can be explored with the Maximal Information Coefficient (MIC); and categorical variables that are dependent to the target class are selected using Gini gain in random forests. Nonetheless, because dependency measures are estimated on finite samples, the interpretability of their quantification and the accuracy when ranking dependencies become challenging. Dependency estimates are not equal to 0 when variables are independent, cannot be compared if computed on different sample size, and they are inflated by chance on variables with more categories. In this paper, we propose a framework to adjust dependency measure estimates on finite samples. Our adjustments, which are simple and applicable to any dependency measure, are helpful in improving interpretability when quantifying dependency and in improving accuracy on the task of ranking dependencies. In particular, we demonstrate that our approach enhances the interpretability of MIC when used as a proxy for the amount of noise between variables, and to gain accuracy when ranking variables during the splitting procedure in random forests.

Adjusted for chance measures are widely used to compare partitions/clusterings of the same data set. In particular, the Adjusted Rand Index (ARI) based on pair-counting, and the Adjusted Mutual Information (AMI) based on Shannon information theory are very popular in the clustering community. Nonetheless it is an open problem as to what are the best application scenarios for each measure and guidelines in the literature for their usage are sparse, with the result that users often resort to using both. Generalized Information Theoretic (IT) measures based on the Tsallis entropy have been shown to link pair- counting and Shannon IT measures. In this paper, we aim to bridge the gap between adjustment of measures based on pair- counting and measures based on information theory. We solve the key technical challenge of analytically computing the expected value and variance of generalized IT measures. This allows us to propose adjustments of generalized IT measures, which reduce to well known adjusted clustering comparison measures as special cases. Using the theory of generalized IT measures, we are able to propose the following guidelines for using ARI and AMI as external validation indices: ARI should be used when the reference clustering has large equal sized clusters; AMI should be used when the reference clustering is unbalanced and there exist small clusters.

There have been a large number of external validity indices proposed for cluster validity. One such class of cluster comparison indices is the information theoretic measures, due to their strong mathematical foundation and their ability to detect non-linear relationships. However, they are devised for evaluating crisp (hard) partitions. In this paper, we generalize eight information theoretic crisp indices to soft clusterings, so that they can be used with partitions of any type (i.e., crisp or soft, with soft including fuzzy, probabilistic and possibilistic cases). We present experimental results to demonstrate the effectiveness of the generalized information theoretic indices.

Mutual information is a very popular measure for comparing clusterings. Previous work has shown that it is beneficial to make an adjustment for chance to this measure, by subtracting an expected value and normalizing via an upper bound. This yields the constant baseline property that enhances intuitiveness. In this paper, we argue that a further type of statistical adjustment for the mutual information is also beneficial - an adjustment to correct selection bias. This type of adjustment is useful when carrying out many clustering comparisons, to select one or more preferred clusterings. It reduces the tendency for the mutual information to choose clustering solutions i) with more clusters, or ii) induced on fewer data points, when compared to a reference one. We term our new adjusted measure the *standardized mutual information*. It requires computation of the variance of mutual information under a hypergeometric model of randomness, which is technically challenging. We derive an analytical formula for this variance and analyze its complexity. We then experimentally assess how our new measure can address selection bias and also increase interpretability. We recommend using the standardized mutual information when making multiple clustering comparisons in situations where the number of records is small compared to the number of clusters considered.

Most current mutual information (MI) based feature selection techniques are greedy in nature thus are prone to sub-optimal decisions. Potential performance improvements could be gained by systematically posing MI-based feature selection as a global optimization problem. A rare attempt at providing a global solution for the MI-based feature selection is the recently proposed Quadratic Programming Feature Selection (QPFS) approach. We point out that the QPFS formulation faces several non-trivial issues, in particular, how to properly treat feature `self-redundancy' while ensuring the convexity of the objective function. In this paper, we take a systematic approach to the problem of global MI-based feature selection. We show how the resulting NP-hard global optimization problem could be efficiently approximately solved via spectral relaxation and semi-definite programming techniques. We experimentally demonstrate the efficiency and effectiveness of these novel feature selection frameworks.