While it possesses many desirable structural properties, the estimation of high-dimensional MI from samples suffers from the curse of dimensionality.

Mutual information (MI) is a fundamental measure of statistical dependence, with a myriad of applications to information theory, statistics, and machine learning. While it possesses many desirable structural properties, the estimation of high-dimensional MI from samples suffers from the curse of dimensionality. Motivated by statistical scalability to high dimensions, this paper proposes sliced MI (SMI) as a surrogate measure of dependence. SMI is defined as an average of MI terms between one-dimensional random projections. We show that it preserves many of the structural properties of classic MI, while gaining scalable computation and efficient estimation from samples. Furthermore, and in contrast to classic MI, SMI can grow as a result of deterministic transformations. This enables leveraging SMI for feature extraction by optimizing it over processing functions of raw data to identify useful representations thereof. Our theory is supported by numerical studies of independence testing and feature extraction, which demonstrate the potential gains SMI offers over classic MI for high-dimensional inference.

Sliced Mutual Information (SMI) is widely used as a scalable alternative to mutual information for measuring non-linear statistical dependence. Despite its advantages, such as faster convergence, robustness to high dimensionality, and nullification only under statistical independence, we demonstrate that SMI is highly susceptible to data manipulation and exhibits counterintuitive behavior. Through extensive benchmarking and theoretical analysis, we show that SMI saturates easily, fails to detect increases in statistical dependence, prioritizes redundancy over informative content, and in some cases, performs worse than correlation coefficient.

While it possesses many desirable structural properties, the estimation of high-dimensional MI from samples suffers from the curse of dimensionality.

Modern neural networks often activate all neurons for every input, leading to unnecessary computation and inefficiency. We introduce Matrix-Interpolated Dropout Layer (MID-L), a novel module that dynamically selects and activates only the most informative neurons by interpolating between two transformation paths via a learned, input-dependent gating vector. Unlike conventional dropout or static sparsity methods, MID-L employs a differentiable Top-k masking strategy, enabling per-input adaptive computation while maintaining end-to-end differentiability. MID-L is model-agnostic and integrates seamlessly into existing architectures. Extensive experiments on six benchmarks, including MNIST, CIFAR-10, CIFAR-100, SVHN, UCI Adult, and IMDB, show that MID-L achieves up to average 55\% reduction in active neurons, 1.7$\times$ FLOPs savings, and maintains or exceeds baseline accuracy. We further validate the informativeness and selectivity of the learned neurons via Sliced Mutual Information (SMI) and observe improved robustness under overfitting and noisy data conditions. Additionally, MID-L demonstrates favorable inference latency and memory usage profiles, making it suitable for both research exploration and deployment on compute-constrained systems. These results position MID-L as a general-purpose, plug-and-play dynamic computation layer, bridging the gap between dropout regularization and efficient inference.

Mutual information (MI) is a fundamental measure of statistical dependence, with a myriad of applications to information theory, statistics, and machine learning. While it possesses many desirable structural properties, the estimation of high-dimensional MI from samples suffers from the curse of dimensionality. Motivated by statistical scalability to high dimensions, this paper proposes sliced MI (SMI) as a surrogate measure of dependence. SMI is defined as an average of MI terms between one-dimensional random projections. We show that it preserves many of the structural properties of classic MI, while gaining scalable computation and efficient estimation from samples.