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Title: Variational lower bounds on mutual information based on nonextensive statistical mechanics

Paper content:
This paper aims to address the limitations of mutual information estimators based on variational optimization.
By redefining the cost using generalized functions from nonextensive statistical mechanics we raise the upper bound of previous estimators and enable the control of the bias variance trade off.
Variational based estimators outperform previous methods especially in high dependence high dimensional scenarios found in machine learning setups.
Despite their performance, these estimators either exhibit a high variance or are upper bounded by log(batch size).
Our approach inspired by nonextensive statistical mechanics uses different generalizations for the logarithm and the exponential in the partition function.
This enables the estimator to capture changes in mutual information over a wider range of dimensions and correlations of the input variables whereas previous estimators saturate them.
Understanding the relationship between two variables is a fundamental problem in machine learning, finance, signal processing and other fields.
To quantify such a relationship, we use mutual information, which measures the mutual dependence between two variables.
The mutual information I(X, Y ) represents the ratio of two probabilities that can account for the nonlinear dependence and is defined as follows:
It is a major challenge to estimate the mutual information in practical scenarios that involve limited samples without knowing the distributions or higher-order statistics McAllester & Statos (2018) ; Paninski (2003) .
For instance, existing methods such as the k-NN based Kraskov et al. (2004) and its variations Gao et al. (2015) ; Wang et al. (2009); Lord et al. (2018) , or KDE based Khan et al. (2007) ; Suzuki et al. (2008) calculate the mutual information by estimating the probability density from the available samples.
Although these approaches perform well in the low dimensional and low dependence case, they do not scale well when either the dimension of the variables or the dependence between variables increases.
Such scenarios are often encountered in machine learning setups.
Estimators based on variational bounds, Belghazi et al. (2018) ; Poole et al. (2018) ; Nguyen et al. (2010) ; ; Zhang (2007); Foster & Grassberger (2011) , perform much better in this scenarios.
These estimators are inspired by the Donsker & Varadhan (1983) representation which states that there exists a function from the sample space to real number that satisfies the following equality: I(X, Y ) = sup f :Ω→R E p(x,y) [f (x, y)] − log E p(y) e f (x,y)
Estimators based on variational bounds replace the function f in the above equation with a neural network trained to maximize a lower bound of the mutual information.
The training process terminates when the lower bounds exhibit convergence, and these bounds are then interpreted as the estimated mutual information values.
This NN-based approach requires good representations of lower bounds and guaranteed convergence for a wide range of dependence between the input variables which leads to numerous challenges.
Current state-of-the-art estimators, when applied to high dimensional high dependence scenarios, either exhibit a high variance or are bounded by log(K), where K is the batch size.
In this work:
1. We propose new variational lower bounds on the mutual information inspired by methods from nonextensive statistical mechanics.
2. We review generalized versions of the logarithm and exponential function, define a generalized version of the partition function, and use them to control the trade off between variance and bias of the estimator.
3. We outperform previous estimators in capturing the trend when varying the correlation and the dimension of the input variables by using different generalizations for the logarithm and partition function.
The main goal of the current work was to improve the performance of mutual information estimators in the high dependence high dimensional scenarios which are often encountered in machine learning setups.
We reviewed previous variational lower bounds and extended them using generalized logarithm and exponential functions from nonextensive statistical mechanics.
One of the most significant findings to emerge from this study was that we can control the trade off between the bias and the variance of the estimator by independently tuning the generalizations q of the logarithm and partition function.
As a result, we are able to better capture the trend when varying the correlation and dependence of the input variables.
This method greatly improves upon the I NCE estimator which has a low variance but a high bias and I α estimator which requires two critics that can sometimes be challenging to train.
The major limitation of the proposed estimator is that its results are not equal in value with the classical mutual information due to use of generalization.
Despite that, I NES still captures the dependence between the input variable and can be applicable to machine learning problems where a mutual information estimator is needed such as feature selection, variational autoencoders, and generative adversarial networks.
Addition and subtraction operations in q-algebra are defined as follow:

assistant: text: This paper tries to establish novel variational lower bounds for mutual information by introducing parameter q and defining q-algebra, showing that the lower bounds have smaller variance and achieves high values.