Computing the Fourier transform of a $q$-ary function $f:\mathbb{Z}_{q}^n\rightarrow \mathbb{R}$, which maps $q$-ary sequences to real numbers, is an important problem in mathematics with wide-ranging applications in biology, signal processing, and machine learning. Previous studies have shown that, under the sparsity assumption, the Fourier transform can be computed efficiently using fast and sample-efficient algorithms. However, in many practical settings, the function is defined over a more general space -- the space of generalized $q$-ary sequences $\mathbb{Z}_{q_1} \times \mathbb{Z}_{q_2} \times \cdots \times \mathbb{Z}_{q_n}$ -- where each $\mathbb{Z}_{q_i}$ corresponds to integers modulo $q_i$. A naive approach involves setting $q=\max_i{q_i}$ and treating the function as $q$-ary, which results in heavy computational overheads. Herein, we develop GFast, an algorithm that computes the $S$-sparse Fourier transform of $f$ with a sample complexity of $O(Sn)$, computational complexity of $O(Sn \log N)$, and a failure probability that approaches zero as $N=\prod_{i=1}^n q_i \rightarrow \infty$ with $S = N^\delta$ for some $0 \leq \delta < 1$. In the presence of noise, we further demonstrate that a robust version of GFast computes the transform with a sample complexity of $O(Sn^2)$ and computational complexity of $O(Sn^2 \log N)$ under the same high probability guarantees. Using large-scale synthetic experiments, we demonstrate that GFast computes the sparse Fourier transform of generalized $q$-ary functions using $16\times$ fewer samples and running $8\times$ faster than existing algorithms. In real-world protein fitness datasets, GFast explains the predictive interactions of a neural network with $>25\%$ smaller normalized mean-squared error compared to existing algorithms.

With the rapid growth of large-scale machine learning models in genomics, Shapley values have emerged as a popular method for model explanations due to their theoretical guarantees. While Shapley values explain model predictions locally for an individual input query sequence, extracting biological knowledge requires global explanation across thousands of input sequences. This demands exponential model evaluations per sequence, resulting in significant computational cost and carbon footprint. Herein, we develop SHAP zero, a method that estimates Shapley values and interactions with a near-zero marginal cost for future queried sequences after paying a one-time fee for model sketching. SHAP zero achieves this by establishing a surprisingly underexplored connection between the Shapley values and interactions and the Fourier transform of the model. Explaining two genomic models, one trained to predict guide RNA binding and the other to predict DNA repair outcome, we demonstrate that SHAP zero achieves orders of magnitude reduction in amortized computational cost compared to state-of-the-art algorithms, revealing almost all predictive motifs -- a finding previously inaccessible due to the combinatorial space of possible interactions.

Protein language models leverage evolutionary information to perform state-of-the-art 3D structure and zero-shot variant prediction. Yet, extracting and explaining all the mutational interactions that govern model predictions remains difficult as it requires querying the entire amino acid space for $n$ sites using $20^n$ sequences, which is computationally expensive even for moderate values of $n$ (e.g., $n\sim10$). Although approaches to lower the sample complexity exist, they often limit the interpretability of the model to just single and pairwise interactions. Recently, computationally scalable algorithms relying on the assumption of sparsity in the Fourier domain have emerged to learn interactions from experimental data. However, extracting interactions from language models poses unique challenges: it's unclear if sparsity is always present or if it is the only metric needed to assess the utility of Fourier algorithms. Herein, we develop a framework to do a systematic Fourier analysis of the protein language model ESM2 applied on three proteins-green fluorescent protein (GFP), tumor protein P53 (TP53), and G domain B1 (GB1)-across various sites for 228 experiments. We demonstrate that ESM2 is dominated by three regions in the sparsity-ruggedness plane, two of which are better suited for sparse Fourier transforms. Validations on two sample proteins demonstrate recovery of all interactions with $R^2=0.72$ in the more sparse region and $R^2=0.66$ in the more dense region, using only 7 million out of $20^{10}\sim10^{13}$ ESM2 samples, reducing the computational time by a staggering factor of 15,000. All codes and data are available on our GitHub repository https://github.com/amirgroup-codes/InteractionRecovery.