# FDSML: Formula-Driven Supervised Metric Learning with Parameter-Aware Triplet Loss

**Authors:** Hina Otake, Shinei Arakawa, Keitaro Tanaka, Yoshihiro Fukuhara, Hirokatsu Kataoka, Shigeo Morishima  
**Lab:** Waseda University · **Published:** 28 Oct 2026 · **Upvotes:** 2  
**Links:** [arXiv](https://arxiv.org/abs/2610.39423) · [PDF](https://arxiv.org/pdf/2610.39423) · [This page](https://omen-bit.github.io/ai-paper-daily/paper/2610.39423.html)

## Key numbers

- **65.6% → 74.0%**: Recall@1 improves from 65.6% to 74.0% on Stanford Online Products and from 56.2% to 78.4% on…
- **56.2% → 78.4%**: …to 74.0% on Stanford Online Products and from 56.2% to 78.4% on Stanford Cars over FDSL…

## Abstract

Formula-Driven Supervised Learning (FDSL) enables vision pre-training without real images, but existing methods cast pre-training as classification on discretized labels, discarding continuity in the generative process. We propose Formula-Driven Supervised Metric Learning (FDSML), which derives relational supervision from image-generation parameters beyond discrete class labels. FDSML combines FineFractalDB, our refined fractal dataset with higher intra-class variation and more stable structures, with a Parameter-Aware Triplet (PA-Triplet) Loss that assigns positives and negatives according to distances in the underlying Iterated Function System parameter space. This objective trains embeddings to respect parameter-derived relative similarity orderings. Experiments show that FDSML improves transferability to image retrieval while remaining competitive on downstream classification and improving robustness under several corruption settings. On ResNet-50, Recall@1 improves from 65.6% to 74.0% on Stanford Online Products and from 56.2% to 78.4% on Stanford Cars over FDSL pre-training. These results suggest that parameter-derived relational supervision provides a stronger synthetic pre-training signal than label-based classification for transfer beyond recognition.

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