# Distances between finite-horizon linear behaviors

arXiv: https://arxiv.org/abs/2503.22849 · HTML: https://mlsanity.com/paper/2503.22849/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Mar 2025.

## Built on by (most cited first, 1 of 1)

- [Robust Least-Squares Optimization for Data-Driven Predictive Control: A Geometric Approach](https://mlsanity.com/paper/2511.09242/) (2025)

## Abstract (arXiv)

The paper introduces a class of distances for linear behaviors over finite time horizons. These distances allow for comparisons between finite-horizon linear behaviors represented by matrices of possibly different dimensions. They remain invariant under coordinate changes, rotations, and permutations, ensuring independence from input-output partitions. Moreover, they naturally encode complexity-misfit trade-offs for Linear Time-Invariant (LTI) behaviors, providing a principled solution to a longstanding puzzle in behavioral systems theory. The resulting framework characterizes modeling as a minimum distance problem, identifying the Most Powerful Unfalsified Model (MPUM) as optimal among all systems unfalsified by a given dataset. Finally, we illustrate the value of these metrics in a time series anomaly detection task, where their finer resolution yields superior performance over existing distances.