# Multi-dimensional concept discovery (MCD): A unifying framework with completeness guarantees

arXiv: https://arxiv.org/abs/2301.11911 · HTML: https://mlsanity.com/paper/2301.11911/

## Key facts

- Cited by 50 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 58 references that are in the map.
- First posted on arXiv 27 Jan 2023.

## Builds on (most cited first, 30 of 38)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Swin Transformer: Hierarchical Vision Transformer using Shifted Windows](https://mlsanity.com/paper/2103.14030/) (2021)
- [Emerging Properties in Self-Supervised Vision Transformers](https://mlsanity.com/paper/2104.14294/) (2021)
- [Grad-CAM: Visual Explanations from Deep Networks via Gradient-based Localization](https://mlsanity.com/paper/1610.02391/) (2016)
- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- ["Why Should I Trust You?": Explaining the Predictions of Any Classifier](https://mlsanity.com/paper/1602.04938/) (2016)
- [Axiomatic Attribution for Deep Networks](https://mlsanity.com/paper/1703.01365/) (2017)
- [Learning Deep Features for Discriminative Localization](https://mlsanity.com/paper/1512.04150/) (2015)
- [Interpretability Beyond Feature Attribution: Quantitative Testing with Concept Activation Vectors (TCAV)](https://mlsanity.com/paper/1711.11279/) (2017)
- [Concept Bottleneck Models](https://mlsanity.com/paper/2007.04612/) (2020)
- [This Looks Like That: Deep Learning for Interpretable Image Recognition](https://mlsanity.com/paper/1806.10574/) (2018)
- [Network Dissection: Quantifying Interpretability of Deep Visual Representations](https://mlsanity.com/paper/1704.05796/) (2017)
- [Evaluating the visualization of what a Deep Neural Network has learned](https://mlsanity.com/paper/1509.06321/) (2015)
- [ResNet strikes back: An improved training procedure in timm](https://mlsanity.com/paper/2110.00476/) (2021)
- [Unmasking Clever Hans Predictors and Assessing What Machines Really Learn](https://mlsanity.com/paper/1902.10178/) (2019)
- [Methods for Interpreting and Understanding Deep Neural Networks](https://mlsanity.com/paper/1706.07979/) (2017)
- [On Completeness-aware Concept-Based Explanations in Deep Neural Networks](https://mlsanity.com/paper/1910.07969/) (2019)
- [Concept Whitening for Interpretable Image Recognition](https://mlsanity.com/paper/2002.01650/) (2020)
- [Explaining by Removing: A Unified Framework for Model Explanation](https://mlsanity.com/paper/2011.14878/) (2020)
- [Net2Vec: Quantifying and Explaining how Concepts are Encoded by Filters in Deep Neural Networks](https://mlsanity.com/paper/1801.03454/) (2018)
- [Acquisition of Chess Knowledge in AlphaZero](https://mlsanity.com/paper/2111.09259/) (2021)
- [Invertible Concept-based Explanations for CNN Models with Non-negative Concept Activation Vectors](https://mlsanity.com/paper/2006.15417/) (2020)
- [GlanceNets: Interpretabile, Leak-proof Concept-based Models](https://mlsanity.com/paper/2205.15612/) (2022)
- [Scikit-dimension: a Python package for intrinsic dimension estimation](https://mlsanity.com/paper/2109.02596/) (2021)
- [Concept Embedding Models: Beyond the Accuracy-Explainability Trade-Off](https://mlsanity.com/paper/2209.09056/) (2022)
- [Concept Activation Regions: A Generalized Framework For Concept-Based Explanations](https://mlsanity.com/paper/2209.11222/) (2022)
- [Neural Basis Models for Interpretability](https://mlsanity.com/paper/2205.14120/) (2022)
- [Scalable Sparse Subspace Clustering by Orthogonal Matching Pursuit](https://mlsanity.com/paper/1507.01238/) (2015)
- [Oracle Based Active Set Algorithm for Scalable Elastic Net Subspace Clustering](https://mlsanity.com/paper/1605.02633/) (2016)
- [Resolving challenges in deep learning-based analyses of histopathological images using explanation methods](https://mlsanity.com/paper/1908.06943/) (2019)

## Built on by (most cited first, 30 of 50)

- [A Holistic Approach to Unifying Automatic Concept Extraction and Concept Importance Estimation](https://mlsanity.com/paper/2306.07304/) (2023)
- [Concept-based Explainable Artificial Intelligence: A Survey](https://mlsanity.com/paper/2312.12936/) (2023)
- [Archetypal SAE: Adaptive and Stable Dictionary Learning for Concept Extraction in Large Vision Models](https://mlsanity.com/paper/2502.12892/) (2025)
- [Universal Sparse Autoencoders: Interpretable Cross-Model Concept Alignment](https://mlsanity.com/paper/2502.03714/) (2025)
- [A Concept-Based Explainability Framework for Large Multimodal Models](https://mlsanity.com/paper/2406.08074/) (2024)
- [Labeling Neural Representations with Inverse Recognition](https://mlsanity.com/paper/2311.13594/) (2023)
- [PURE: Turning Polysemantic Neurons Into Pure Features by Identifying Relevant Circuits](https://mlsanity.com/paper/2404.06453/) (2024)
- [Decoupling Pixel Flipping and Occlusion Strategy for Consistent XAI Benchmarks](https://mlsanity.com/paper/2401.06654/) (2024)
- [Explainable AI for Time Series via Virtual Inspection Layers](https://mlsanity.com/paper/2303.06365/) (2023)
- [From Colors to Classes: Emergence of Concepts in Vision Transformers](https://mlsanity.com/paper/2503.24071/) (2025)
- [Understanding the (Extra-)Ordinary: Validating Deep Model Decisions with Prototypical Concept-based Explanations](https://mlsanity.com/paper/2311.16681/) (2023)
- [Concept-Based Explanations in Computer Vision: Where Are We and Where Could We Go?](https://mlsanity.com/paper/2409.13456/) (2024)
- [Concept Bottleneck Models Without Predefined Concepts](https://mlsanity.com/paper/2407.03921/) (2024)
- [Explaining Deep Learning for ECG Analysis: Building Blocks for Auditing and Knowledge Discovery](https://mlsanity.com/paper/2305.17043/) (2023)
- [Relational Concept Bottleneck Models](https://mlsanity.com/paper/2308.11991/) (2023)
- [Understanding Visual Feature Reliance through the Lens of Complexity](https://mlsanity.com/paper/2407.06076/) (2024)
- [Beyond Scalars: Concept-Based Alignment Analysis in Vision Transformers](https://mlsanity.com/paper/2412.06639/) (2024)
- [ConSim: Measuring Concept-Based Explanations' Effectiveness with Automated Simulatability](https://mlsanity.com/paper/2501.05855/) (2025)
- [A survey on Concept-based Approaches For Model Improvement](https://mlsanity.com/paper/2403.14566/) (2024)
- [Generating Counterfactual Trajectories with Latent Diffusion Models for Concept Discovery](https://mlsanity.com/paper/2404.10356/) (2024)
- [Using explainable AI to investigate electrocardiogram changes during healthy aging -- from expert features to raw signals](https://mlsanity.com/paper/2310.07463/) (2023)
- [Local Concept Embeddings for Analysis of Concept Distributions in Vision DNN Feature Spaces](https://mlsanity.com/paper/2311.14435/) (2023)
- [Unsupervised Interpretable Basis Extraction for Concept-Based Visual Explanations](https://mlsanity.com/paper/2303.10523/) (2023)
- [Bi-Orthogonal Factor Decomposition for Vision Transformers](https://mlsanity.com/paper/2601.05328/) (2026)
- [Structuring Sparsity: Block-Sparse Featurizers Capture Visual Concept Manifolds](https://mlsanity.com/paper/2606.25234/) (2026)
- [Towards Human-Understandable Multi-Dimensional Concept Discovery](https://mlsanity.com/paper/2503.18629/) (2025)
- [Manipulating Feature Visualizations with Gradient Slingshots](https://mlsanity.com/paper/2401.06122/) (2024)
- [A Geometric Unification of Concept Learning with Concept Cones](https://mlsanity.com/paper/2512.07355/) (2025)
- [Hierarchical Concept-based Interpretable Models](https://mlsanity.com/paper/2602.23947/) (2026)
- [Efficient and Flexible Neural Network Training through Layer-wise Feedback Propagation](https://mlsanity.com/paper/2308.12053/) (2023)

## Abstract (arXiv)

The completeness axiom renders the explanation of a post-hoc XAI method only locally faithful to the model, i.e. for a single decision. For the trustworthy application of XAI, in particular for high-stake decisions, a more global model understanding is required. Recently, concept-based methods have been proposed, which are however not guaranteed to be bound to the actual model reasoning. To circumvent this problem, we propose Multi-dimensional Concept Discovery (MCD) as an extension of previous approaches that fulfills a completeness relation on the level of concepts. Our method starts from general linear subspaces as concepts and does neither require reinforcing concept interpretability nor re-training of model parts. We propose sparse subspace clustering to discover improved concepts and fully leverage the potential of multi-dimensional subspaces. MCD offers two complementary analysis tools for concepts in input space: (1) concept activation maps, that show where a concept is expressed within a sample, allowing for concept characterization through prototypical samples, and (2) concept relevance heatmaps, that decompose the model decision into concept contributions. Both tools together enable a detailed understanding of the model reasoning, which is guaranteed to relate to the model via a completeness relation. This paves the way towards more trustworthy concept-based XAI. We empirically demonstrate the superiority of MCD against more constrained concept definitions.