Neural Network Interpretability

Susceptibilities are a technique for neural network interpretability that studies the response of posterior expectation values of observables to perturbations of the loss. We generalize this construction to the setting of the regret in deep reinforcement learning and investigate the utility of susceptibilities in a simple gridworld model that nevertheless exhibits non-trivial stagewise development. We argue that susceptibilities reveal internal features of the development of the model in parameter space that one cannot detect purely by studying the development of the learned policy. We validate these results with activation-steering, and discuss the framework's extension to RLHF post-training.

These notes introduce the theory of susceptibilities as developed in [arXiv:2504.18274, arXiv:2601.12703] for interpreting neural networks. The susceptibility of an observable $φ$ to a data perturbation is defined as a derivative of a posterior expectation, which by the fluctuation--dissipation theorem equals a posterior covariance. Different choices of $φ$ yield different objects: per-sample losses give the influence matrix (the Bayesian influence function of [arXiv:2509.26544]), while component-localized observables give the structural susceptibility matrix that pairs model components with data patterns. The susceptibility matrix is (up to a factor of $nβ$) the Jacobian of the map from data distributions to structural coordinates; its pseudo-inverse provides a linearized solution to the patterning problem of [arXiv:2601.13548]: finding data perturbations that produce a desired structural change. We motivate the theory from its statistical-mechanical foundations, then give a detailed exposition of susceptibilities, their empirical estimators, and their connection to the geometry of the loss landscape.

Implicit neural representations (INRs) encode images as neural-network weights, making image classification a problem of weight-space classifiability. A natural geometric hypothesis is that classifier feedback should make image-specific weights cluster by class in the shared-anchor coordinate. We test this hypothesis in the SIREN-based Meta Weight Transformer (MWT) regime, where end-to-end training meta-learns a shared initialization and inner-loop update schedule for fitting image-specific SIRENs. We find that this prediction fails. Exposed weight-space geometry and supervised clustering pressure do not reliably track trained-reader accuracy; clustering can even make local neighborhoods more class-consistent while making the trained reader worse. Crucially, the reader constructs rather than inherits class-aligned geometry: token-flow diagnostics show that class-aligned neighborhoods become strongly predictive of trained-reader accuracy only after late reader interactions, not in the input coordinate. We further identify the native SIREN bias column in the augmented weight token as a low-dimensional, sample-dependent causal readout route for the trained reader; targeted controls rule out generic scalar-column and marginal-distribution artifacts. The diagnosis motivates interventions that strengthen reader routing, add an explicit bias route, or use denser inner-loop fitting; under the lane-specific training conventions used here, route-directed variants often outperform the shared-anchor baseline but interact non-additively. Task-induced INR weights are classifiable not because they form raw geometric clusters, but because their class signal is routed through the reader.

Concept-based explanations offer a promising approach for explaining the predictions of deep neural networks in terms of high-level, human-understandable concepts. However, existing methods either do not establish a causal connection between the concepts and model predictions or are limited in expressivity and only able to infer causal explanations involving single concepts. At the same time, the parallel line of work on formal abductive and contrastive explanations computes the minimal set of input features causally relevant for model outcomes but only considers low-level features such as pixels. Merging these two threads, in this work, we propose the notion of concept-based abductive and contrastive explanations that capture the minimal sets of high-level concepts causally relevant for model outcomes. We then present a family of algorithms that enumerate all minimal explanations while using concept erasure procedures to establish causal relationships. By appropriately aggregating such explanations, we are not only able to understand model predictions on individual images but also on collections of images where the model exhibits a user-specified, common behavior. We evaluate our approach on multiple models, datasets, and behaviors, and demonstrate its effectiveness in computing helpful, user-friendly explanations.

Concept Bottleneck Models (CBMs) have become a popular approach to enable interpretability in neural networks by constraining classifier inputs to a set of human-understandable concepts. While effective, current models embed concepts in flat Euclidean space, treating them as independent, orthogonal dimensions. Concepts, however, are highly structured and organized in semantic hierarchies. To resolve this mismatch, we propose Hyperbolic Concept Bottleneck Models (HypCBM), a post-hoc framework that grounds the bottleneck in this structure by reformulating concept activation as asymmetric geometric containment in hyperbolic space. Rather than treating entailment cones as a pre-training penalty, we show they encode a natural test-time activation signal: the margin of inclusion within a concept's entailment cone yields sparse, hierarchy-aware activations without any additional supervision or learned modules. We further introduce an adaptive scaling law for hierarchically faithful interventions, propagating user corrections coherently through the concept tree. Empirically, HypCBM rivals post-hoc Euclidean models trained on 20$\times$ more data in sparse regimes required for human interpretability, with stronger hierarchical consistency and improved robustness to input corruptions.

How can an arbitrary attribution method be generalized from first principles to capture interactions? We answer this with the metagame, a conceptual framework for quantifying second-order interaction effects of model explanations. We cast the attribution value $φ_i$ of feature $i$ as a cooperative game among the other features and compute its Shapley value, which measures how much feature $j$ influences the attribution of $i$, yielding the directional meta-attribution $\varphi_{j \to i}$. By decomposing attribution itself rather than the model directly, meta-attributions extend any gradient- or attention-based method to interactions, uniting removal-based perturbations with model internals. Theoretically, we prove that meta-attributions sum to the first-order attribution they explain, a hierarchical decomposition that Shapley interactions and integrated Hessians turn out to perform implicitly. Empirically, we demonstrate that meta-attributions deliver insights across diverse interpretability applications: (i) quantifying token interactions in instruction-tuned language models, (ii) explaining cross-modal similarity in vision-language encoders, and (iii) interpreting text-to-image concepts in multimodal diffusion transformers.

Piecewise affine neural networks (PANNs) provide a principled geometric perspective on neural network expressivity by characterizing the input--output map as a continuous piecewise affine (CPA) function whose complexity is governed by the number, arrangement, and shapes of its affine regions. However, existing interpretability and expressivity analyses often rely on indirect proxies (e.g., activation statistics or theoretical upper bounds) and rarely offer practical, accurate tools for enumerating and visualizing the induced region partition under realistic architectures and bounded input domains. In this work, we present AffineLens, a unified framework for computing the hyperplane arrangements and polyhedral structures underlying PANNs. Given a calibrated (bounded) input polytope, AffineLens identifies the subset of neuron-induced hyperplanes that intersect the domain, enumerates the resulting affine sub-regions in a layer-wise manner, and returns provably non-empty maximal CPA regions together with interior representatives. The framework further provides visualizations of region partitioning and decision boundaries, enabling qualitative inspection alongside quantitative region counts. By exploiting the affine restriction property of CPA networks under fixed activation patterns, AffineLens supports a broad class of modern components, including batch normalization, pooling, residual connections, multilayer perceptrons, and convolutional layers. Finally, we use AffineLens to perform a systematic empirical study of architectural expressivity, comparing networks through region complexity metrics and revealing how design choices influence the geometry of learned functions.

Unlike MLPs, Kolmogorov-Arnold Networks (KANs) expose explicit learnable edge functions on every connection, enabling mechanistic explanation in time-series forecasting. This paper introduces Temporal Functional Circuits, a framework that transforms KAN edge functions from latent visualizations into faithful, temporally grounded explanations. Built on a gated residual KAN that decomposes forecasts into a linear base and a sparsely activated KAN correction, the framework (i) maps each edge to input lags via output-aware attribution, (ii) ranks edges by learned activation range, and (iii) validates faithfulness through edge-level interventions including zeroing and spline removal. Removing the learned B-spline component while retaining the base SiLU term degrades forecasts, providing evidence that the spline shape itself carries predictive value beyond the base activation. On four synthetic regimes of increasing complexity, the learned gate opens progressively wider as signal complexity grows. On regime-switching signals, gated KAN achieves 59% lower MSE than linear-only models. Across eight benchmarks, the gated architecture is competitive with linear, attention, and MLP alternatives, while providing interpretable edge functions that MLP-based corrections cannot offer.

Training loss and throughput can hide distinct internal representation in language-model training. To examine these hidden mechanics, we use spectral measurements as practical and operational diagnostics. Using a controlled family of decoder-only models adapted from the modded NanoGPT codebase, we introduce an empirical protocol based on activation covariance and per-sample gradient SVD spectra. This dual-view reveals three empirical findings and one mechanistic explanation. First, batch size acts as a latent determinant of representation geometry: runs that reach equal loss settle into systematically distinct activation spectra. Second, the activation covariance tail measured early in training reliably forecasts downstream token efficiency. Third, movement of the activation spectrum head (leading modes), together with gradient spectra, characterizes underlying learning-dynamics changes, separating learning-side architectural improvements from primarily execution-side gains. These predictive and diagnostic signals persist across the 12-, 36-, and 48-layer model tiers. Finally, a mechanistic model proves the main observations and explains how activation covariance spectra correlate with task-aligned feature learning.

Understanding the neural mechanisms underlying visual computation has long been a central challenge in neuroscience. Recent alignment based approaches have improved the accuracy of decoding visual stimuli from brain activity, yet they provide limited insight into the neural computations that give rise to these improvements. To address this gap, we propose Dual-Tower Image-Neural Alignment (DINA), an interpretable contrastive framework for analyzing population level visual computations in primary visual cortex (V1). DINA jointly trains a biologically motivated dual-tower architecture that aligns visual stimuli and corresponding V1 population responses in a shared latent space at the level of intermediate feature maps, enabling both accurate decoding and direct access to interpretable feature maps. Evaluated on large-scale two-photon calcium imaging data from mouse V1, DINA achieves accurate neural-based decoding while revealing that decoding performance is primarily supported by coarse, low-level visual structure, rather than semantic category information or fine-grained details. Further analysis reveals that alignable feature maps emerge from multiple spatially distributed image regions, capturing both shape and texture cues, and are predominantly reconstructed by sparse subsets of strongly responsive neurons and their functional interactions. Together, these results confirm that, beyond enabling accurate decoding, DINA provides a principled framework for probing the computational mechanisms underlying visual processing in V1.

Kolmogorov-Arnold networks (KANs) offer a potent combination of accuracy and interpretability, thanks to their compositions of learnable univariate activation functions. However, the activations of well-fitting KANs tend to exhibit pathologically high-curvature oscillations, making them difficult to interpret, and standard regularization penalties do not prevent this. Here we derive a basis-agnostic curvature penalty and show that penalized models can maintain accuracy while achieving substantially smoother activations. Accounting for how function composition shapes curvature, we prove an upper bound on the full model's curvature relative to the curvature penalty, and use this to motivate richer forms of penalties. Scientific machine learning is increasingly bottlenecked by the trade-off between accuracy and interpretability. Results such as ours that improve interpretability without sacrificing accuracy will further strengthen KANs as a practical tool for both prediction and insight.

Understanding how neural models represent human-interpretable concepts is challenging. Prior work has explored linear concept subspaces from diverse perspectives, such as probing and concept erasure. We introduce a unified framework to study these subspaces along two axes: \textit{containment}, which tests if a concept is fully represented in a subspace but not outside it, and \textit{disentanglement}, which tests for isolation from other concepts. In experiments on both text and speech models, we first highlight that concept subspaces may not be uniquely determined, and discuss the implications for concept subspace analysis. Then, we compare properties of concept subspaces estimated using five estimators, proposed in different communities. We find that (1) the choice of estimator impacts the containment and disentanglement properties; (2) the state-of-the-art concept erasure method, LEACE, performs well on both testing axes, but still struggles to generalize to unseen data; and (3) in HuBERT speech representations, phone information is both contained and disentangled from speaker information, while speaker information is hard to contain in a compact subspace, despite being disentangled from phones.

We introduce HyCOP, a modular framework that learns parametric PDE solution operators by composing simple modules (advection, diffusion, learned closures, boundary handling) in a query-conditioned way. Rather than learning a monolithic map, HyCOP learns a policy over short programs - which module to apply and for how long - conditioned on regime features and state statistics. Modules may be numerical sub-solvers or learned components, enabling hybrid surrogates evaluated at arbitrary query times without autoregressive rollout. Across diverse PDE benchmarks, HyCOP produces interpretable programs, delivers order-of-magnitude OOD improvements over monolithic neural operators, and supports modular transfer through dictionary updates (e.g., boundary swaps, residual enrichment). Our theory characterizes expressivity and gives an error decomposition that separates composition error from module error and doubles as a process-level diagnostic.

Modern vision models achieve remarkable accuracy, but explaining where evidence arises, what the model encodes, and how internal computations assemble that evidence remains fragmented. We introduce an iERF-centric framework that unifies local, global, and mechanistic interpretability around a single analysis unit: the pointwise feature vector (PFV) paired with its instance-specific Effective Receptive Field (iERF). On the local side, Sharing Ratio Decomposition (SRD) expresses each PFV as a mixture of upstream PFVs via sharing ratios and propagates iERFs to construct class-discriminative saliency maps. SRD yields high-resolution, activation-faithful explanations, is robust to targeted manipulation and noise, and remains activation-agnostic across common nonlinearities. For the global view, we introduce Concept-Anchored Feature Explanation (CAFE), which utilizes the iERF as a semantic label, grounding abstract latent vectors in verifiable pixel-level evidence. With CAFE, we address the challenge of non-localized sparse autoencoder latents--especially in Transformers, where early self-attention mixes distant context. To answer how representations are composed through depth, we propose the Interlayer Concept Graph with Interlayer Concept Attribution (ICAT), which quantifies concept-to-concept influence while isolating layer pairs; an interlayer insertion, deletion protocol identifies Integrated Gradients as the most faithful instantiation. Empirically, across ResNet50, VGG16, and ViTs, our framework outperforms baselines in both fidelity and robustness, successfully interprets dispersed SAE features, and exposes dominant concept routes in correct, misclassified, and adversarial cases. Grounded in iERFs, our approach provides a coherent, evidence-backed map from pixels to concepts to decisions.

Sparse autoencoders (SAEs) are widely used to extract interpretable features from neural network representations, often under the implicit assumption that concepts correspond to independent linear directions. However, a growing body of evidence suggests that many concepts are instead organized along low-dimensional manifolds encoding continuous geometric relationships. This raises three basic questions: what does it mean for an SAE to capture a manifold, when do existing SAE architectures do so, and how? We develop a theoretical framework that answers these questions and show that SAEs can capture manifolds in two fundamentally different ways: globally, by allocating a compact group of atoms whose linear span contains the entire manifold, or locally, by distributing it across features that each selectively tile a restricted region of the underlying geometry. Empirically, we find that SAEs suboptimally recover continuous structures, mixing the global subspace and local tiling solutions in a fragmented regime we call dilution. This explains why manifold structure is rarely visible at the level of individual concepts and motivates post-hoc unsupervised discovery methods that search for coherent groups of atoms rather than isolated directions. More broadly, our results suggest that future representation learning methods should treat geometric objects, not just individual directions, as the basic units of interpretability.

A foundational assumption in CNN interpretability -- that deep encoders suppress background pixels while classifiers merely select from a cleaned feature pool (the Spatial Funnel Hypothesis) -- remains untested due to spatial hallucinations in existing visualization tools. We address this by introducing a hallucination-free inversion framework built on magnitude-phase decoupling and Local Adjoint Correctors. Our method mathematically guarantees that the spatial gradient support of every reconstruction stems strictly from genuinely active channels. Using this framework as a geometric probe, we uncover the first pixel-level evidence of strong superposition in vision encoders. We show that per-channel inversions are uniformly holographic: positive and negative weight reconstructions are visually and energetically indistinguishable. However, their algebraic sum sharply concentrates on the foreground. This proves classification operates via destructive interference -- classifier weights cancel a shared background direction in pixel space and constructively assemble class-discriminative residuals, directly falsifying the Spatial Funnel Hypothesis. This interference model identifies the volume of the admissible interference subspace as the geometric quantity governing channel requirements. We prove this volume is dual to the GAP covariance determinant, yielding a covariance-volume channel selection algorithm with a $(1-1/e)$ approximation guarantee. This algorithm mathematically reveals out-of-distribution (OOD) failure as a measurable collapse of the covariance volume essential for interference-based classification. Our framework extends seamlessly to attention-based heads without retraining.

Deep learning models are used in critical applications, in which mistakes can have serious consequences. Therefore, it is crucial to understand how and why models generate predictions. This understanding provides useful information to check whether the model is learning the right patterns, detect biases in the data, improve model design, and build systems that can be trusted. This work proposes a new method for interpreting Convolutional Neural Networks in image classification tasks. The approach works by selecting the most representative feature maps that contribute to each prediction. To solve this combinatorial problem, we encode it into a quantum constrained optimization problem and propose to solve it using quantum annealing. We evaluate our method against the state-of-the-art explainable AI techniques, specifically GradCAM and GradCAM++, and observe an improved class disentanglement, i.e. the model's decision boundaries become more distinct and its reasoning more transparent. This demonstrates that our approach enhances the quality of explanations, making it easier to understand which features the model relies on for specific predictions. In addition, we study the computational behavior of the quantum annealing algorithm. Specifically, we analyze the minimum energy gap of the system during computation and the probability that the algorithm finds the correct solution. These analyses provide theoretical insight into why the method works effectively in practice.

Gradient-based saliency methods are widely used to interpret deep neural networks, yet they often produce noisy and unstable explanations that poorly align with semantically meaningful input features. We argue that a fundamental cause of this behavior lies in the geometry of learned representations: correlated feature dimensions diffuse attribution gradients across redundant directions, resulting in blurred and unreliable saliency maps. To address this issue, we identify feature correlation as a structural limitation of gradient-based interpretability and propose SaliencyDecor, a training framework that enforces feature decorrelation to improve attribution fidelity without modifying saliency methods or model architectures by reshaping the feature space toward orthogonality, our approach promotes more concentrated gradient flow and improves the fidelity of saliency-based explanations. SaliencyDecor jointly optimizes classification, prediction consistency under feature masking, and a decorrelation regularizer, requiring no architectural changes or inference-time overhead. Extensive experiments across multiple benchmarks and architectures demonstrate that our method produces substantially sharper and more object-focused saliency maps while simultaneously improving predictive performance, achieving accuracy gains across the datasets. These results establish our method as a principled mechanism for enhancing both interpretability and accuracy, challenging the conventional trade-off between explanation quality and model performance.

We investigate the explanability properties of the recently proposed linear-min-max neural networks. At initialization, they can be interpreted as k-medoids with the infinity norm as a distance. Then, they are trained using subgradient descent to better fit the data. The model has been shown to be a universal approximator. Yet, we can trace the decision process because a single most activated neuron is responsible for the value of the output. Using this property, we designed a pixel fragility measure that determines whether changes to a single pixel may be responsible to a change in the classification output. Experiments on the PneumoniaMnist dataset show that this explanation for the output of the neural network compares favorably to SHAP and Integrated Gradient.

Neural networks can be trained to learn task-relevant representations from data. Understanding how these networks make decisions falls within the Explainable AI (XAI) domain. This paper proposes to study an XAI topic: analysing, visualising and understanding the unknown organisation of network representations, particularly those a speaker recognition network learns from utterances, for recognising speaker identity. Past studies have employed algorithms (e.g. K-means) to analyse the different ways in which network representations can be naturally grouped into clusters, i.e. to analyse different flat clustering phenomena within the space defined by those representations. In contrast, this work applies two algorithms -- Single-Linkage Clustering (SLINK) and Hierarchical Density-Based Spatial Clustering of Applications with Noise (HDBSCAN) -- to analyse the different ways in which representations from the speaker recognition network can form clusters with hierarchical relationships, i.e., to analyse different hierarchical clustering phenomena within the representation space of the speaker recognition network. Furthermore, an algorithm called Hierarchical Cluster-Class Matching (HCCM) is designed to semantically interpret one of the above hierarchical clustering phenomena analysed using SLINK. Given the clusters representing this phenomenon, HCCM identifies which ones best match individual semantic classes related to gender and nationality (e.g.\ male, female, Ireland, UK) and and-logic conjunctions of these classes (e.g.\ female and Ireland). The Liebig score metric is also proposed within HCCM to quantify the matching quality of each cluster-class pair and diagnose the factor that limits each match.

As the demand to integrate Artificial Intelligence into high-stakes environments continues to grow, explaining the reasoning behind neural-network predictions has shifted from a theoretical curiosity to a strict operational requirement. Our work is motivated by the explanations of autoregressive neural predictions on dynamic physical fields, as in weather forecasting. Gradient-based feature attribution methods are widely used to explain the predictions on such data, in particular due to their scalability to high-dimensional inputs. It is also interesting to remark that gradient-based techniques such as SmoothGrad are now standard on images to robustify the explanations using pointwise averages of the attribution maps obtained from several noised inputs. Our goal is to efficiently adapt this aggregation strategy to dynamic physical fields. To do so, our first contribution is to identify a fundamental failure mode when averaging perturbed attribution maps on dynamic physical fields: stochastic input perturbations do not induce stationary amplitude noise in attribution maps, but instead cause a geometric displacement of the attributions. Consequently, pointwise averaging blurs these spatially misaligned features. To tackle this issue, we introduce WassersteinGrad, which extracts a geometric consensus of perturbed attribution maps by computing their entropic Wasserstein barycenter. The results, obtained on regional weather data and a meteorologist-validated neural model, demonstrate promising explainability properties of WassersteinGrad over gradient-based baselines across both single-step and autoregressive forecasting settings.

Neuron labeling assigns textual descriptions to internal units of deep networks. Existing approaches typically rely on highly activating examples, often yielding broad or misleading labels by focusing on dominant but incidental visual factors. Prior work such as FALCON introduced contrastive examples -- inputs that are semantically similar to activating examples but elicit low activations -- to sharpen explanations, but it primarily addresses subspace-level interpretability rather than scalable neuron-level labeling. We revisit contrastive explanations for neuron-level labeling in two stages: (1) candidate label generation with vision language models (VLMs) and (2) label assignment with CLIP-like encoders. First, we show that providing contrastive image sets to VLMs yields candidate labels that are more specific and more faithful. Second, we introduce Contrastive Semantic Projection (CSP), an extension of SemanticLens that incorporates contrastive examples directly into its CLIP-based scoring and selection pipeline. Across extensive experiments and a case study on melanoma detection, contrastive labeling improves both faithfulness and semantic granularity over state-of-the-art baselines. Our results demonstrate that contrastive examples are a simple yet powerful and currently underutilized component of neuron labeling and analysis pipelines.

Feature attribution methods explain the predictions of deep neural networks by assigning importance scores to individual input features. However, most existing methods focus solely on marginal effects, overlooking feature interactions, where groups of features jointly influence model output. Such interactions are especially important in image classification tasks, where semantic meaning often arises from pixel interdependencies rather than isolated features. Existing interaction-based methods for images are either coarse (e.g., superpixel-only) or, fail to satisfy core interpretability axioms. In this work, we introduce H-Sets, a novel two-stage framework for discovering and attributing higher-order feature interactions in image classifiers. First, we detect locally interacting pairs via input Hessians and recursively merge them into semantically coherent sets; segmentation from Segment Anything (SAM) is used as a spatial grouping prior but can be replaced by other segmentations. Second, we attribute each set with IDG-Vis, a set-level extension of Integrated Directional Gradients that integrates directional gradients along pixel-space paths and aggregates them with Harsanyi dividends. While Hessians introduce additional compute at the detection stage, this targeted cost consistently yields saliency maps that are sparser and more faithful. Evaluations across VGG, ResNet, DenseNet and MobileNet models on ImageNet and CUB datasets show that H-Sets generate more interpretable and faithful saliency maps compared to existing methods.

Several brain foundation models (FM) have recently been proposed to predict brain disorders by modelling dynamic functional connectivity (FC). While they demonstrate remarkable model performance and zero- or few-shot generalization, the salient features identified as potential biomarkers are yet to be thoroughly evaluated. We propose RE-CONFIRM, a framework for evaluating the robustness of potential biomarker candidates elucidated by deep learning (DL) models including FMs. From experiments on five large datasets of Autism Spectrum Disorder (ASD), Attention-deficit Hyperactivity Disorder (ADHD), and Alzheimer's Disease (AD), we found that although commonly used performance metrics provide an intuitive assessment of model predictions, they are insufficient for evaluating the robustness of biomarkers identified by these models. RE-CONFIRM metrics revealed that simply finetuning FMs leads to models that fail to capture regional hubs effectively, even in disorders where hubs are known to be implicated, such as ASD and ADHD. In view of this, we propose Hub-LoRA (Low-Rank Adaptation) as a fine-tuning technique that enables FMs to not only outperform customised DL models but also produce neurobiologically faithful biomarkers supported by meta-analyses. RE-CONFIRM is generalizable and can be easily applied to ascertain the robustness of DL models trained on functional MRI datasets. Code is available at: https://github.com/SCSE-Biomedical-Computing-Group/RE-CONFIRM.

We establish a mathematical correspondence between state space models, a state-of-the-art architecture for capturing long-range dependencies in data, and an exactly solvable nonlinear oscillator network. As a specific example of this general correspondence, we analyze the diagonal linear time-invariant implementation of the Structured State Space Sequence model (S4). The correspondence embeds S4D, a specific implementation of S4, into a ring network topology, in which recent inputs are encoded, as waves of activity traveling over the one-dimensional spatial layout of the network. We then derive an exact operator expression for the full forward pass of S4D, yielding an analytical characterization of its complete input-output map. This expression reveals that the nonlinear decoder in the system induces interactions between these information-carrying waves that enable classifying real-world sequences. These results generalize across modern SSM architectures, and show that they admit an exact mathematical description with a clear physical interpretation. These insights enable a new level of interpretability for these systems in terms of nonlinear oscillator networks.

The analysis of neural representation has become an integral part of research aiming to better understand the inner workings of neural networks. While there are many different approaches to investigate neural representations, an important line of research has focused on doing so through the lens of intrinsic dimensions (IDs). Although this perspective has provided valuable insights and stimulated substantial follow-up research, important limitations of this approach have remained largely unaddressed. In this paper, we highlight a crucial discrepancy between theory and practice of IDs in neural representations, theoretically and empirically showing that common ID estimators are, in fact, not tracking the true underlying ID of the representation. We contrast this negative result with an investigation of the underlying factors that may drive commonly reported ID-related results on neural representation in the literature. Building on these insights, we offer a new perspective on ID estimation in neural representations.

Multimodal Large Language Models (MLLMs) effectively integrate text and audio to interpret context in complex interactive dialogues. However, the internal mechanisms by which heterogeneous modalities influence model behavior remain opaque. While Shapley Values (SV) provide a robust, model-agnostic framework for local explainability in text-based NLP, their extension to multimodal data is hindered by cross-channel dependencies, intricate dialogue structures, and the prohibitive computational complexity of dense audio representations. In this work, we formalize a multimodal extension of the Shapley Value framework, treating discrete text tokens and aligned audio segments as cooperative features. To ensure computational feasibility, we deploy a suite of efficient estimation strategies: exact SV computation for low-dimensional inputs and sampling-based approximations - including Monte Carlo permutations and stratified sampling with Neyman-optimal allocation - to minimize variance under constrained computational budgets. To resolve the granularity mismatch between modalities, we propose Spectrogram-Guided Phonetic Alignment (SGPA), a novel preprocessing method that maps high-frequency audio streams to interpretable, word-aligned segments. Our contribution is twofold: first, we provide an open-source, model-agnostic Python package and a companion GUI for the computation and interactive visualization of multimodal attributions. Second, we evaluate our framework using curated subsets of the VoiceBench and Infinity Instruct datasets across diverse multilingual scenarios. Our experimental results reveal that input modality is a primary driver of attribution volatility and demonstrate that standard syntactic importance proxies often fail to predict model attention in multimodal, cross-lingual contexts.

Vision Transformers (ViTs) have become a dominant architecture in computer vision, yet their prediction process remains difficult to interpret because information is propagated through complex interactions across layers and attention heads. Existing attention based explanation methods provide an intuitive way to trace information flow. However, they rely mainly on raw attention weights, which do not explicitly reflect the final decision and often lead to explanations with limited class discriminability. In contrast, gradient based localization methods are more effective at highlighting class specific evidence, but they do not fully exploit the hierarchical attention propagation mechanism of transformers. To address this limitation, we propose Decision-Aware Attention Propagation (DAP), an attribution method that injects decision-relevant priors into transformer attention propagation. By estimating token importance through gradient based localization and integrating it into layer wise attention rollout, the method captures both the structural flow of attention and the evidence most relevant to the final prediction. Consequently, DAP produces attribution maps that are more class sensitive, compact, and faithful than those generated by conventional attention based methods. Extensive experiments across Vision Transformer variants of different model scales show that DAP consistently outperforms existing baselines in both quantitative metrics and qualitative visualizations, indicating that decision aware propagation is an effective direction for improving ViT interpretability.

AlphaGeometry represents a milestone in neuro-symbolic reasoning, yet its architecture faces a log-linear scaling bottleneck within its symbolic deduction engine that limits its efficiency as problem complexity increases. Recent technical reports suggest that current domain-specific languages may be isomorphic as input representations to natural language, interchanging them acts as a performance-invariant transformation, implying that current neural guidance relies on superficial encodings rather than structural understanding. This paper addresses this representation bottleneck by proposing a logic-to-topology encoding designed to reveal the structural invariants of a model's latent space under a transformation of its input space. By leveraging the Logic of Observation, we utilize the duality between provability in observable theories and topologies to propose a logic-to-topology encoder for the input space. We introduce the concept of the "topological dual of a dataset", a transformation that bridges formal logic, topology, and neural processing. This framework serves as a Rosetta Stone for neuro-symbolic AI, providing a principled pathway for the mechanistic interpretability of how models navigate complex discovery paths.

In this article, we establish a precise connection between binarized neural networks (BNNs) and Sugeno integrals. The advantage of the Sugeno integral is that it provides a framework for representing the importance of inputs and their interactions, while being equivalent to a set of if-then rules. For a hidden BNN neuron at inference time, we show that the activation threshold test can be written as a Sugeno integral on binary inputs. This yields an explicit set-function representation of each neuron decision, and an associated rule-based representation. We also provide a Sugeno-integral expression for the last-layer score. Finally, we discuss how the same framework can be adapted to support richer input interactions and how it can be extended beyond the binary case induced by binarized neural networks.