# Resolving Multi-Modal Regression by Difference-Quotient-Based Clustering:Fast Coarse Conditional-Label Assignment

arXiv: https://arxiv.org/abs/2608.25467 · HTML: https://mlsanity.com/paper/2608.25467/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 7 references that are in the map.
- First posted on arXiv 26 Aug 2026.

## Builds on (most cited first, 4 of 4)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Learning in an Uncertain World: Representing Ambiguity Through Multiple Hypotheses](https://mlsanity.com/paper/1612.00197/) (2016)
- [Overcoming Limitations of Mixture Density Networks: A Sampling and Fitting Framework for Multimodal Future Prediction](https://mlsanity.com/paper/1906.03631/) (2019)

## Abstract (arXiv)

Multimodal regression suffers from the mean-collapse pathology: under squared loss, an unconstrained regressor converges to the conditional mean, which for K > 1 lies away from all modes. We attribute this failure to pairwise contradictions--samples with nearly identical inputs but distant outputs--and propose Difference-Quotient Clustering (DQC), which partitions data to minimize intra-cluster output-vs-input discrepancy. Each sample is assigned to the cluster that minimizes its maximum contradiction ratio; a logits generator and a conditional network are then trained on the resulting labels. Since the generating modality is unknown at test time, we evaluate via minimum squared error (minMSE) against all K true outputs. On synthetic benchmarks (K=5, 10), DQC achieves test minMSE 0.19 (K=5, nx=500), versus 0.09 for an oracle, 1.08 for random labels, and 1.33 for mean collapse. We observe two empirical regularities: larger intra-cluster contradictions require deeper networks, and oracle labels generalize from fewer samples than cluster-derived equivalents. The clustering is a hard, parallelizable O(n^2/2) front-end for coarse conditional assignment, reducing the burden of downstream generative refinement. A second-stage re-clustering on residual errors is outlined as future work.