End-to-End MI Maximization

• End-to-end mutual information maximization is a framework that optimizes all components of a differentiable system by maximizing statistical dependence between inputs and outputs.
• It leverages neural MI estimators, score-based gradients, and kernel methods to address the intractability of mutual information computation in high-dimensional spaces.
• This approach enhances applications in representation learning, multimodal fusion, and semantic communications through comprehensive, joint optimization of complex architectures.

End-to-end mutual information maximization refers to learning system parameters so as to directly maximize the mutual information (MI) between an input and output across an entire complex network, typically with all components differentiable and trainable simultaneously. MI quantifies the statistical dependence between variables and is central to unsupervised, supervised, and multimodal learning, as well as communication and sensing system optimization. End-to-end maximization frameworks utilize neural MI estimators, gradient-based surrogates, and differentiable pipelines to optimize MI objectives, often in the presence of architectural constraints and high-dimensional data.

1. Foundational Principles and Objectives

The central aim of end-to-end MI maximization is to learn model parameters that maximize $I(X; Y)$ or conditional MI $I(Y; R | S)$, where $X$ is the system input, $Y$ is output (or task-relevant variable), and $S$ may be auxiliary side information such as channel state. Formally,

$$I(X; Y) \equiv h(Y) - h(Y|X) = \mathbb{E}_{p(x, y)} \left[ \log\frac{p(x, y)}{p(x)p(y)} \right]$$

This principle extends to:

• Representation learning: maximizing $I(\text{features}; \text{labels})$ (Ozdenizci et al., 2021)
• Communication systems: maximizing $I(\text{codeword}; \text{channel output})$ (Fritschek et al., 2019)
• Multimodal fusion: maximizing $I(\text{image features}; \text{text features})$ or their local MI variants (Liao et al., 2021)
• Complex DAG systems: maximizing MI across input–output paths (Wadayama, 5 Jan 2026)
• Semantic communications: maximizing conditional MI for classification post channel (Cai et al., 2024)

End-to-end MI maximization is distinguished by global (joint) optimization across a system, often including encoders, decoders, discriminators, and surrogates.

2. Mutual Information Estimators and Gradients

Exact MI computation is generally intractable in high dimensions. Dominant approaches employ:

• Neural MI Estimators: Donsker–Varadhan bound (MINE), Jensen–Shannon, InfoNCE, contrastive predictive coding (CPC), and f-divergence duals. For example, MINE utilizes (Fritschek et al., 2019):

$$I_\theta(X; Y) \geq \mathbb{E}_{p(x, y)}\left[ T_\theta(x, y) \right] - \log \mathbb{E}_{p(x)p(y)}\left[ e^{T_\theta(x, y)} \right]$$

where $T_\theta$ is a critic network.

• Score-Based Gradients: For DAGs, MI gradients with respect to network parameters $\theta$ use score functions:

$$\nabla_\theta I(X; Y) = \mathbb{E}[ (D_\theta Y)^T (s_{Y|X}(Y|X) - s_Y(Y)) ]$$

with $s_{Y|X}(y|x) = \nabla_y \log p_{Y|X}(y|x)$ and efficient computation via vector–Jacobian products (VJP) (Wadayama, 5 Jan 2026).

• Nonparametric KDE Estimation: MMINet directly estimates MI gradients via kernel density estimators without parametric assumptions (Ozdenizci et al., 2021).
• Bayesian Nonparametric Estimators: Finite Dirichlet Process (DP) approximations regularize MINE, reducing gradient variance and improving stability (Fazeliasl et al., 11 Mar 2025).

3. End-to-End Optimization Frameworks

End-to-end MI maximization is architected as joint training pipelines:

• Multimodal Learning: Simultaneously optimizing image encoder, text encoder, and MI discriminators with local or global objectives, selecting maximal MI pairs (Liao et al., 2021).
• Graph Representation Learning: Multi-view MI (feature, topology), reconstruction and diversity regularization are optimized jointly; all submodules receive gradients from MI estimators in a unified loss (Fan et al., 2021).
• Communication Systems: Channel encoder parameters are updated to maximize neural MI estimates, independent of differentiable channel models (Fritschek et al., 2019).
• Autoencoders: InfoMax variants maximize $I(\text{input}; \text{latent})$ by latent entropy estimation and reconstruction error (Crescimanna et al., 2019), or via explicit MI terms in VAE loss (Rezaabad et al., 2019).

Typical workflow involves:

• Forward pass: encode data, calculate MI estimates via neural or nonparametric critics
• Backpropagation: update model parameters jointly to increase MI lower bounds (or minimize surrogate losses)
• Stabilization: use model-specific regularization (disagreement, DP smoothing), data augmentation, and batch negative sampling

4. Surrogates, Constraints, and Calibration

In complex systems, direct optimization of MI with respect to design parameters may be non-differentiable or costly. Surrogate models are introduced:

• Local Surrogate Optimization: In detector design, batch-evaluated MI values (via MINE) are fit by a local neural surrogate, whose gradients approximate true MI gradients for end-to-end layer optimization (Wozniak et al., 18 Mar 2025).
• Global Constraints: Projected gradient ascent is employed for MI maximization under cost functions (total power, etc.), with projections performed after each parameter update (Wadayama, 5 Jan 2026).
• Digital Twin Calibration: Fisher divergence minimization aligns the output distribution of a simulation model to real system statistics using only output samples, leveraging score-based MI gradient machinery (Wadayama, 5 Jan 2026).

5. Practical Architectures and Applications

End-to-end MI maximization architectures are diverse:

• ResNet/BERT-based multimodal encoders (Liao et al., 2021)
• Multi-branch GCN encoders for graph views (Fan et al., 2021)
• Fully-connected, deep unfolded networks for communication and domain-driven MIMO precoders (Cai et al., 2024)
• Generative models (autoencoders, GANs) with MI regularization (Crescimanna et al., 2019, Rezaabad et al., 2019, Zuo et al., 2020)

Applications extend to:

• Biomedical dimensionality reduction and classification (Ozdenizci et al., 2021)
• Robust document hashing and discrete representation learning (Stratos et al., 2020)
• Semantic communications and cooperative edge inference (Cai et al., 2024)
• High energy physics detector geometry optimization (Wozniak et al., 18 Mar 2025)
• Multi-modal pretraining for vision-language, dense reconstruction, and unified supervised/self-supervised pipelines (Su et al., 2022)
• Multimodal image-to-image translation, yielding enhanced diversity via explicit MI regularization (Zuo et al., 2020)

A representative summary of methods is given in the table below.

System Type	Key MI Maximization Mechanism	Core Citation
Multimodal Fusion	Local MI + neural discriminators	(Liao et al., 2021)
Graph Representation	JS-bound MINE + multi-view objective	(Fan et al., 2021)
Communication/Detection	DV-bound MINE + surrogate regression	(Fritschek et al., 2019, Wozniak et al., 18 Mar 2025)
Dimensionality Reduction	Nonparametric KDE – MI gradient	(Ozdenizci et al., 2021)
Autoencoder/Generative	Latent entropy + reconstruction loss	(Crescimanna et al., 2019, Rezaabad et al., 2019)
Semantic Communication	Conditional MI + deep unfolded nets	(Cai et al., 2024)
BNP Estimator (General)	DP-regularized neural MI	(Fazeliasl et al., 11 Mar 2025)

6. Theoretical Guarantees and Empirical Performance

Rigorous MI estimators and surrogate designs yield sufficient conditions for unbiased estimation and stable convergence:

• Finite-sample consistency: Bayesian nonparametric estimators converge almost surely to true MI; DP-based regularization is strictly tighter on expectation than empirical MINE (Fazeliasl et al., 11 Mar 2025)
• Gradient estimation accuracy: Score-matching MI gradients in DAGs reliably reproduce analytic ground truth across linear and nonlinear scenarios (Wadayama, 5 Jan 2026)
• Representation robustness: InfoMax approaches avoid posterior collapse and yield higher active units, discriminative latents, and improved downstream classification (Rezaabad et al., 2019)

Empirical studies reveal:

• Locally maximized MI in multimodal fusion surpasses global MI and supervised-only baselines in downstream metrics (AUC, cluster separability) (Liao et al., 2021)
• Surrogate-based MI optimization in detector design matches physics-informed approaches yet enables task-agnostic tuning (Wozniak et al., 18 Mar 2025)
• Multi-view MI learning in graph embeddings gives consistent state-of-the-art unsupervised classification and clustering results (Fan et al., 2021)
• Deep unfolded precoding architectures for MIMO semantic comms provide rapid, robust E2E accuracy improvements over LMMSE baselines (Cai et al., 2024)
• DP-regularized MI estimation reduces batch sensitivity, accelerates generative model convergence, and improves perceptual match metrics (Fazeliasl et al., 11 Mar 2025)

7. Extensions, Limitations, and Future Directions

End-to-end MI maximization frameworks are extensible to federated learning, reinforcement learning (policy MI), and digital twin calibration. Limitations include batch size requirements for neural MI stability, curse of dimensionality in nonparametric estimators, and dependence on expressive surrogates for true MI gradients.

Potential extensions:

• Adaptive MI estimators balancing bias and variance via DP, kNN, or hybrid approaches (Fazeliasl et al., 11 Mar 2025)
• Task-specific constraint integration, e.g., resource budgets, semantic fidelity (Wadayama, 5 Jan 2026, Cai et al., 2024)
• Unified multi-modal pretraining pipelines with conditional and joint MI lower bounds (Su et al., 2022)
• Score-based divergence objectives for unsupervised calibration of complex physical systems (Wadayama, 5 Jan 2026)

End-to-end mutual information maximization thus provides a principled, theoretically grounded, and practically flexible basis for joint learning, optimization, and calibration of complex systems in diverse high-dimensional domains.