LUMA: Ultralight Low-Rank Forecasting for Periodic Time Series

Long-term Time Series Forecasting (LTSF) faces a growing dichotomy between parameter-heavy foundation models and lightweight linear models. While recent trends favor massive scale, in this paper, we investigate whether the spectral compactness of periodic signals can motivate efficient low-rank forecasting. By combining period-folding with a rank-constrained operator, we reduce parameter growth through a compact cross-period representation. We introduce **LUMA** (**L**ow-rank **U**ltralight **M**atrix **A**pproximation), a minimalist architecture designed to exploit this efficiency. LUMA couples period-folding with a rank-constrained operator to 1) preserve within-period positions in a cross-period representation and 2) forecast their evolution through shared low-rank factors. We further prove that for a $K$-harmonic signal, the period-folded representation has rank at most $2K$, providing a structural motivation for rank-constrained forecasting. Extensive experiments on seven benchmarks demonstrate that LUMA achieves the best MSE on four datasets (ETTh1, ETTh2, ETTm2, Weather) while remaining competitive on the rest. In the Electricity profiling configuration, LUMA uses only **635 parameters**, approximately $13,000\times$ fewer than PatchTST and $36\times$ fewer than TimeKAN. Relative to SparseTSF, this configuration reduces parameters by approximately **36%** and MACs by approximately **18%**, with a reported CPU inference latency of **1.3 ms**. These results support low-rank cross-period modeling as an effective approach to accurate forecasting with a small parameter budget.