SHAP zero Explains Biological Sequence Models with Near-zero Marginal Cost for Future Queries
Authors: Darin Tsui
Aryan Musharaf
Yigit Efe Erginbas
Justin Singh Kang
Amirali Aghazadeh
Venue: NeurIPS 2025 poster ()
Keywords: Fourier transform
Möbius transform
Shapley values
Shapley interactions
Biological sequences
Genomics
Proteins
Primary area: social_and_economic_aspects_of_machine_learning
TL;DR: SHAP zero estimates Shapley values and interactions with a near-zero marginal cost for future queried sequences through the Fourier transform.

Abstract: The growing adoption of machine learning models for biological sequences has intensified the need for interpretable predictions, with Shapley values emerging as a theoretically grounded standard for model explanation. While effective for local explanations of individual input sequences, scaling Shapley-based interpretability to extract global biological insights requires evaluating thousands of sequences—incurring exponential computational cost per query. We introduce SHAP zero, a novel algorithm that amortizes the cost of Shapley value computation across large-scale biological datasets. After a one-time model sketching step, SHAP zero enables near-zero marginal cost for future queries by uncovering an underexplored connection between Shapley values, high-order feature interactions, and the sparse Fourier transform of the model. Applied to models of guide RNA efficacy, DNA repair outcomes, and protein fitness, SHAP zero explains predictions orders of magnitude faster than existing methods, recovering rich combinatorial interactions previously inaccessible at scale. This work opens the door to principled, efficient, and scalable interpretability for black-box sequence models in biology.

SHAP zero Explains Biological Sequence Models with Near-zero Marginal Cost for Future Queries
Darin Tsui Georgia Institute of Technology darint gatech.edu
Aryan Musharaf
Yigit Efe Erginbas
Georgia Institute of Technology
UC Berkeley erginbas @ berkeley.edu
amusharaf3 gatech.edu
Justin Singh Kang
Amirali Aghazadeh
UC Berkeley justin kang berkeley.edu
Georgia Institute of Technology amiralia @ gatcch.eatttt
Abstract
The growing adoption of machine learning models for biological sequences has intensified the need for interpretable predictions, with Shapley values emerging theoretically grounded standard for model explanation. While effective for local explanations of individual input sequences, scaling Shapley-basee inter- pretability to extract global biological insights requires evaluating thousands of requences-incune
exponential computational cost per query. We introduce SHAP zero, novel algorithm that amortizes the cost of Shapley value computa- tion across large-scale biological datasets. After one-time model sketching step, marginal cost for future queries by uncovering underexplored connection between Shapley values, high-order feature interactions, and the sparse Fourier transform of the model. Applied to models of guide RNA efficacy, DNA repair outcomes, and protein fitness, SHAP zero explains predictions orders of magnitude faster than existing methods, recovering rich combinatorial in- teractions previously inaccessible at scale. This work opens the door to principled, efficient, and scalable interpretability for black-box sequence models in biology.
SHAP zero enables
Introduction
The remarkable success of machine learning modeling biological sequences, from DNA to proteins, has created an urgent need for interpretability tools that can reveal what these models have learned. Black-box sequence models now guide genome editing (T3),, protein design [466, and regulatory variant prediction (749). yet explaining their predictions remains prohibitively expensive at scale [10].
In this work, we study the problem of explaining black-box models that map a length-n biological sequence to a real-valued prediction. We denote such models as f Z - R. where each sequence E Z consists of symbols from an alphabet of size 9 (e.g., for DNA, goal is to explain the predictions of f' over Q many input sequences X1 ....... XQ at scale. A popular interpretability framework is SHapley Additive exPlanations (SHAP) [I], which assigns an additive importance score 1SV (j) to each input feature j in a given sequence These scores are grounded in cooperative game theory 112 and quantify the marginal contribution of feature j (111-111))) (erug)(x;) ur (x:)].
for proteins). Our
across all subsets of other features: 15V (i) = =EEEEEA)
2 where D (1,2,...,,,) n is the full feature set. The value function vr(x;) denotes the pected output when only features in are fixed and the rest are marginalized over: ur(x;) =
T Σ Emmppz!! _ fmm. See Appendix A for more details Shapley values.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Fourier aketching (one-time)
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Figure 1: Overview of SHAP zero. a, SHAP zero pays a one-time cost to create a global Fourier sketch of f. This illustration shows 4 Fourier coefficients strategically aliased into multiple subsampled transforms (U1, U2), and recovered by identifying singleton bins. b, For each future query X1.. xg. SHAP zero localizes the global sketch via the Möbius transform of order é, capturing query-specifer feature interactions. This maps to c, SHAP values, and d, Shapley interactions. By marginalizing the cost of future queries, SHAP zero enables e, scalable amortized explanations and f, discovery of biological motifs at unprecedented scale.
Unfortunately, computing exact SHAP values requires evaluating f on an exponential number of perturbed sequences, making intractable even for modest n. Stochastic estimators [I] [3 such as KernelSHAP [I]] reduce the cost with random sampling, while model-based approximators [I] 19|24] exploit internal structure but are often restricted to white-box access. Recent algorithms for
Shapley interaction indices (95) such SHAP-IQ (25,, further increase this burden, requiring even more model evaluations.
These challenges are particularly severe in interpretation of biological sequence models, where one often seeks explanations across thousands of query sequences in a dataset to gain a global insight about the underlying biological phenomenon. For instance, estimating SHAP values for about thousand guide RNA sequences in the gene editing model TIGER took one day on our single NVIDIA RTX A6000 machine. Estimating third-order Faithful Shapley (Faith-Shap) interactions through SHAP-IQ is projected to take over 80 days highlightgn a fundamental scalability bottleneck.
This raises a key question: can we amortize the cost of explaining sequence models across queries?
A Fourier view of SHAP. We observe that model evaluations used to compute Shapley explanations for one sequence contain information that can be reused to explain other sequences. This motivates a different paradigm: rather than estimating SHAP values and interactions de novo for each sequence, we can first sketch the blxck-box model globally, then compute local explanations from this sketch.
To realize this, we uncover powerful and underexplored connection between Shapley explanations and the model's Fourier transform through the Möbius transform. The Fourier basis provides a global, query-agnostic decomposition of f. and if f is compressible in this basis-as many biological sequence models are 32 we can estimate its top-s coefficients efficiently. To obtain localized feature contributions, we introduce novel mapping from the Fourier transform to the localized Möbius transform around each query, reorganizing the global sketch to isolate interacting features. This global-to-lock framework enables efficient amortized computation of Shapley explanations across future queries without recomputing model evaluations, and, as a result, allows for the discovery of high-order features (motifs) at unprecedented scales (Fig. Ip.
Contributions. We present SHAP zero, new algorithm for efficient model explanation of black-box sequence models via sparse Fourier sketching. After initial one-time sketching phase, SHAP zero estimates SHAP values and Shapley interactions with near-zero additional cost per query sequence.
Our key contributions are:
Fourier sketching for biological sequences. We develop sample-efficient algorithm to recover the top-s Fourier coefficients of f, defined over Zž, with sample complexity (sn?) and runtime O (sn²), that is, only polynomial dependence on the input sequence length

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Formal connection between Fourier and SHAP through the Möbius transform. We introduce a formal pipeline for mapping global Fourier coefficients the localized Möbius transform, and then to SHAP values and interactions, enabling principled amortized explanations.
Scalable amortized explanation and motif discovery. We conduct large-scale experiments to explain two genomic models, TIGER [3 and inDelphi [34), and the protein language model Tranception [35] with SHAP zero. We demonstrate that SHAP zero estimates feature interactions with an amortized computational cost up to 1000 times faster than current methods. SHAP zero reveals GC content of the seed region in TIGER, microhomologous motifs inDelphi, and epistatic interactions in Tranception
predictive features, task previously inaccessible due to the combinatorial space of possible feature interactions. Software for SHAP zero available at https:://itthb. com/ waoom/mmiggouuuooo cowes//paaooeesspppppspaa
2 Background
Set Möbius Transform. We begin by first introducing the classic Möbius transform in relationship to SHAP values and interactions. The classic Möbius transform formulation marginalizes the contributions between interactions of sets [36] 37.. However, our global sketch via the Fourier transform captures interactions between features a sequence. For this reason, we will first introduce the classic Möbius transform formulation, which we will refer to as the set Möbius transform for clarity, and introduce the SHAP and Faith-Shap interaction equation in terms of the set Möbius transform. Then, in later section, we will introduce an extension of the Möbius transform for q-ary functions. The definition the set Möbius transform is as follows.
Definition 2.1. (Set Möbius transform). Given a value function v 2" + R and a set S, the set Möbius transform 0x, (v, S) is defined using the forward and inverse transform:
Forward: ax, (v, S) =
(T (T()),
Inverse:
(T())) = Σ
ax ax, (v,
(1)
TES
SCT
Each set Möbius transform coefficient ax; (v, S). where S 7, represents the marginal contribution of the subset S, given a value function in determining the score (x,)). Reconstructing the original score (x;)) from all subsets S C T requires taking a linear sum over all the marginal contributions.
The set Möbius transform naturally provides a bridge between SHAP values (38) and Faith-Shap interactions (2).. We give both equations below, where, given a maximum interaction order f FST (T) is the Faith-Shap interaction index over a set T in a given sequence x;:
(1)
Σ TUUH Ox TU,TU TED:O)
FES x:
0x (v. T)
Value Function in g-ary Functions. The computation of SHAP values and Faith-Shap interactions depends on the value function vT (x,). which quantifies how much the features in T contribute toward the prediction of fxx).. In this paper, we define the value function as the expectation of fx,) with respect to Xyp: (x,,)
Ep(x,r) [xx]). Here, take the expectation over the marginal distribution p(x:7) ) for the missing inputs. Ep(x.p) can be computed for any Shapley problem by approximating the missing inputs x;7- However, in q-ary functions, we can compute this equation exactly because we are constrained to alphabets at each site; every possible missing input must be in z!f By assuming that every possibility of x 2 is equally likely, the value function reduces to taking an average contribution over !!?l possible inputs.
3 The SHAP zero Algorithm
In this section, we detail the SHAP zero algorithm. SHAP zero estimates SHAP values and interac- tions in three steps: (i) Estimating Fourier coefficients, (ii) Computing the Möbius transform, and (iā) Finding SHAP values and Faith-Shap interactions. overview of SHAP zero provided in Algorithms and 2 in Appendix
3

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3.1 Estimating Fourier Coefficients
The first key step in SHAP zero is to compute the sparse Fourier transform of (Fig. function f /:2g - IR can be expressed in terms of its Fourier transform Fly] as:
Ta). Any
f(m)=
Σ yE2g
(ly)
E Zg.
(3)
where w c and y is the frequency vector. The Fourier transform provides global sketch of f. irrespective of the input query sequence.
Computing the Fourier transform exactly requires obtaining q" samples from J. which can be prohibitive for large values of or n. Fortunately, in practice, sequence models tend to have a sparse Fourier transform [30 32 39 40 meaning Fly] only has a few non-zero coefficients.
SHAP zero takes advantage of sparsity and pays a one-time cost to estimate the top-s Fourier coefficients of f. We leverage structured subsampling in If using patterns from sparse-graph codes [40]- 42). which implicitly hash Fourier coefficients into buckets. To do this in sequence models, SHAP zero creates C' many subsampling groups. For each subsampling group c = 1... C, we define random subsampling matrix M. E 2nxe, where b is the subsampling dimension. Additionally, we define a total of P offsets de.p E 24 where E [P]- The subsampled transform [e] indexed by j E is computed as: U..p j] B ElEZ; f(Me + de.p) w
By subsampling f. we leverage a classical result in signal processing, which states that regularly subsampling signal in time introduces predictable aliasing patterns in the Fourier domain. Using M. and de.p: the induced aliasing structure satisfies:
Ueplii
Σ
F[k] w (de...))
MMK=
Thus, each Ue [i] is linear combination of F[k] that are hashed into buckets defined by M.. Our goal is to maximize the number of [j] that contain exactly one non-zero Fourier coefficient, which we call a singleton. Singletons allow direct recovery of the corresponding F[k]. However, inevitably, some U.. j] will contain more than one Fourier coefficient, which we call multi-ton. In Fig. [.. we alias the 4 non-zero Fourier coefficients into two buckets, U1 and U2 (where we drop the notation in Ue.p for simplicity). assuming de,p is vector of all zeros. U1 generated two singletons containing F y1 and Fly4] and one multi-ton, and U2 similarly generated two singletons containing Fly2) and F Y3) and one multi-ton, allowing us to recover the entire Fourier transform.
In practice, though, the Fourier transform may not be able to be fully recovered in shot. Therefore, we can resolve multi-tons to generate more singletons by modeling the aliasing structure across subsampling groups as sparse bipartite graph. SHAP zero then maximizes the number of singletons recovered by applying a peeling procedure 40 Upon recovery of singleton, its contribution can be subtracted from adjacent multi-tons, potentially creating new singletons. This peeling process continues until no further singletons can be found. SHAP zero recovers the top-s Fourier coefficients with a sample complexity of O(sn²) and a computational complexity of O(sn²). Appendix C contains more details on computing the sparse Fourier transform in sequence models.
3.2 Computing the Möbius Transform
The second step of SHAP zero is to map the sparse Fourier coefficients of to the Möbius transform of I around an input query sequence x; (Fig. [b). We build on the binary Möbius transform formulation found in 43 for functions characterized in 2y IR, which we dub the q-ary Möbius transform.
We first define the q-ary Möbius transform a function of f. Then, describe how it can be estimated via the Fourier transform. We denote m < k to be m, = k; or m, = for all of i, and subtraction of k m for m < k by standard real field subtraction. The q-ary Möbius transform around the input sequence x,, Mx, [k), where is the feature interaction vector, is defined as: Definition 3.1. (q-ary Möbius transform). Given Σ + R. the q-ary Möbius transform Mx, [k), where k is the feature interaction vector, can be defined as,
Mx. [k]
Σ (-1)llk-m) mllo f((m + x4) mod g), m, EN
(5)
k

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with its inverse defined as,
f((m + x²) mod q)
Σ Mx, [k]. kSm
(6)
Mx, [k] captures how the output of f changes when features in x; are perturbed (Fig. D The modulo operation is used to ensure the inequalities m S k and k S m hold when x; is encoded as a vector in ž Appendix illustrates the differences in computing the set and g-ary Möbius transform.
Without assumptions, the q-ary Möbius transform scales with (qq") computations, which is in- tractable for large values of or n. In order to bypass these computational issues, we map the top-s Fourier coefficients Mx, [k]. In Appendix E.I, we prove the following Proposition:
Proposition 3.2. (Fourier transform q-ary Möbius transform). Given the top-s Fourier coefficients
Fly) with a maximum order of and the input query sequence Xs, Mx, [k]
is defined as:
Mx. k) = Σ Flyl
(-I)lkk-mloo(mms) (m>))
Vk E Zy.
YE
Imsk
where the compulational complexity of Equation
scales with (()2(2n²))
Here, is the maximum order of Fourier interactions recovered by SHAP zero and, in practice, bounded by & S 5 [32] Since Fly] provides global sketch of the model irrespective of x,, the shifting property of the Fourier transform (see Equation (2T) in Appendix (E.T)) is first used to align Fly] with x, Then, Equation
3 is plugged into Equation and simplified to obtain the final result.
3.3 Finding SHAP Values and Faith-Shap Interactions
The last step of SHAP zero to map Mx, [k] to SHAP values and Faith-Shap interactions. We have a direct relationship going from the set Möbius transform SHAP values and Faith-Shap interactions. Therefore, computing Shapley explanations for g-ary functions first requires a conversion from the q-ary Möbius to the Möbius transform. In Appendix E.2 we prove the following Proposition:
Proposition 3.3. (q-ary Möbius transform to set Möbius transform). Computing the set Möbius transform 0z, (v, S) given Mx, [k] is defined as:
ax (v, S) ()= (-1)
Σ k1ks>0
Mx, [k].
gllkllo
The proof follows two major steps. After computing Mx, [k]. the inverse q-ary Möbius transform from Equation (6) is plugged into the g-ary value function. Then, the g-ary function is plugged into the forward set Möbius transform to obtain Equation {
2). we prove the following Theorem: Theorem 3.4. (q-ary Möbius transform to SHAP values). Given f 2n - R and Mx. [k], the SHAP
Using Proposition 3.3 and the YS (j) equation from Equation
value equation can be written as:
SV SV = Σ
|k/loglikll Mx. [k].
kik,
with computational complexity of Ogg)), where &. is the maximum q-ary Mübius coefficients order.
The proof can be found in Appendix E.3 Here, the > operation is defined over the standard real field in ž and the Lo norm | counts the number of nonzero elements in k (Fig. [o). 1$V (j) can be interpreted as the contribution of the 5th feature the prediction of fxx;); when I$V (j) > substituting the 7th feature to a different feature will, on average, decrease the value of fx;), and vice versa. Therefore, 1SV (j) sums over q-ary Möbius transform coefficients, which capture the marginal effects of all subsets of features on fxx), with negative sign.
Similarly, SHAP zero can map the q-ary Möbius transform to I FS (T) (Fig. 1d). Since the 1FSI (T) equation in Equation (2) is already written in terms of ax (v, S):
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Figure 2: SHAP zero enables scalable amortized explanations in TIGER. a. TIGER 33 predicts the guide score of 26 length target sequences. b, The estimated top Fourier coefficients by SHAP zero outperform linear and pairwise models in predicting the guide scores in a held-out set. c, SHAP value estimates reveal high agreements (p 0.83) between SHAP zero and KerneISHAP. d, Total runtime of SHAP zero against KernelSHAP is marked by x in plots that depict the computational cost versus the number of explained sequences in both algorithms. e, Histogram of Faith-Shap interactions from SHAP zero compared to SHAP-IQ (see Appendix [F). f, Total runtime of SHAP zero versus SHAP-IQ in TIGER demonstrate that SHAP zero is more than 1000-fold faster.
Theorem 3.5. (q-ary Möbius transform to Faith-Shap). Given f ZzZy R and Mx, [k] with maximum order the hh order Faith-Shap interaction index equation can be written as:
(T)
Σ k:
gllkloo Mx[(k)
(10)
with computationnll complexity of Og').
The proof in Appendix E.4 hinges on the fact that the Faith-Shap interaction index and the q-ary
Möbius transform are both 4t order. If this is the case, computing Faith-Shap is equivalent to
converting from the q-ary Möbius transform to the set Möbius transform from Equation (8). Faith-
Shap can be seen as a generalization of the SHAP value for feature interactions [26]: when computing Faith-Shap interactions becomes identical to computing SHAP values using Equation
(
3.4
Computational Complexity of SHAP zero
After paying a one-time cost to find the top-s Fourier coefficients, which scale with poly(n). SHAP zero explanations do not scale with n. By bounding the computational complexity of Proposition 3.2 Theorem and Theorem 3.5 by L, remove the computational dependency on The dependence on f does not impact SHAP zero in practice, since É is typically a small constant for sequence models. Additionally, while Equations 7 (9. and (10) depend on q. in biological sequences, we are restricted to either q 4 nucleotides or q = 20 amino acids. These combined enable SHAP zero to explain new query sequence with near-zero marginal cost (essentially free).
Experiments
Explaining Guide RNA Binding. TIGER 33 is a convolutional neural network trained to predict the binding efficiency CRISPR-Cas 3d guide RNA (gRNA) to a target DNA sequences (Fig. h)) We considered length-n = 26 input sequences to the model that perfectly match the target and guide RNA sequences that reverse complement the target. Positions 1-3 correspond to the additional 5 context given to the target sequence and positions 6- 12 of the target sequence correspond to the

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Figure 3: SHAP zero reveals high-order motif's in inDelphi. a, inDelphi (3] predicts DNA repair outcomes
40 length sequences. b. Recovered Fourier coefficients outperform linear and pairwise models in repair outcomes in a held-out set. AUROC and AUPRC are not reported due to the regression nature of the model (see Appendix G.7). c, SHAP zero and KernelSHAP estimates reveal the importance of nucleotides around the cut site. d, Total runtime of SHAP zero versus KernelSHAP. e. Histogram of Faith Paith-Shap interactions in SHAP zero compared to SHAP-IQ (see Appendix (F). High-order feature interactions identified by SHAP zero reveal the importance of microhomology patterns around the cut site (see Appendix f, Total runtime of SHAP zero versus SHAP-IQ.
short segment that facilitates the initial pairing with the target [44] and is especially sensitive to mutations [45)
nucleotides that bind with the seed region in the gRNA:
To assess the accuracy of the recovered Fourier coefficients from SHAP zero, predict the exper- imental guide scores of 1038 held-out target sequences (Fig. 2p). SHAP zero's 's recovered Fourier coefficients predict experimental guide scores with Pearson correlation of p 0.58, outperforming the linear (p- = 0.44) and pairwise (p- = 0.48) model with L.2 regularization, suggesting that it successfully captures high-order (E > 1) feature interactions. SHAP zero and KernelSHAP estimates e are in high agreements (p- = 0.83), with 47% and 56% of their top positive estimates attributed to C and G nucleotides in the seed region, respectively, highlighting the GC content as a predictive feature (see Fig. 2 and Appendix [F). However, it took us an amortized time of only 6.5 seconds to compute SHAP zero values compared to 109 seconds find KemelSHHA values (Fig. 11) a 17-fold speedup.
We then explained the high-order feature interactions with SHAP zero and compared it with SHAP-IQ. The overlapping interactions in SHAP zero and SHAP-IQ. constituting 95% and 5% of the total interactions, respectively, correlate with = 0.79. We expected most interactions to be located however, observed that only 34% of SHAP-IQ's interactions within the seed region, compared to 52% of SHAP zero's 's interactions (Fig. 2b). Besides, estimating the Faith-Shap interactions took amortized time of 6.5 seconds for SHAP zero compared to hours for SHAP-IQ (Fig. 2), suggesting that SHAP zero is more than 1000-fold faster and enables a biologically feasible explanation of feature interactions.
within the seed region [3] [46;:
Explaining DNA Repair Outcome. inDelphi (34] is a machine learning model trained to predict DNA repair outcomes after CESSPR-Cas--nduuee double-stranded breaks, including the probability of Cas9 inducing a one- or two-base pair insertion or deletion (frameshift frequency) (Fig. We considered length-n 40 input DNA target sequences, where the cut site occurs between positions
20 and 21, region that is considered to have the most predictive power 30 34 47).
We used the recovered Fourier coefficients from SHAP zero to predict the experimental frameshift frequencies of 84 held-out target sequences (Fig. 3b). SHAP zero (p- = 0.74) significantly outper- formed linear (p- = 0.37) and pairwise models cannot be explained with just 2nd order interactions. SHAP zero and KernelSHAP estimates are in high agreement again (p- 0.84) (see Fig. 3 and Appendix F). This time, it took SHAP zero an
0.47), suggesting that inDelphi's interactions
7

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Figure 4: SHAP zero uncovers epistatic interactions in Tranception. a. We analyze the green florescence protein over = 10 epistatic sites from [49] b. SHAP zero Fourier coefficients outperform linear and pairwise models in predicting protein function in a held-out set. c, SHAP zero and KernelSHAP estimates (p 0.97) reveal the importance of secondary structure promoters. d. Total runtime of SHAP zero versus KemelSHAP. e, Heatmap of Faith-Shap interactions run over 200 sequences with SHAP zero and 50 sequences with SHAP-IQ. f, Total runtime of SHAP zero versus SHAP-IQ.
amortized time of 40 minutes to estimate the SHAP values compared to KernelSHAP's 55 minutes (Fig. Bi). The difference in amortized time is smaller here than in TIGER due to the smaller set of query sequences.
We then used SHAP zero and SHAP-IQ to extract high-order feature interactions. The overlapping interactions (constituting 85% and 3% of the total interactions SHAP zero and SHAP-IQ, respec- tively) correlate with / 0.74. However, just 17% of SHAP-IQ's interactions are within three nuclcotides of the cut site compared to 55% of SHAP zero's interactions (Fig. (p.). Similar to TIGER, noticed a significant speedup in amortized time when using SHAP zero compared to SHAP-IQ: SHAP zero took 40 minutes compared to 37 hours from SHAP-IQ (Fig. 3D.
We performed a detailed analysis of the top SHAP zero feature interactions of inDelphi (see Ap- pendix FF. Our analysis reveals repeating motifs around the cut site as key high-order features predictive of DNA repair [30, 34. These high-order features can be attributed to microhomology patterns that mediate long deletions around the cut site (Fig. M 48 Notably, a top feature identified by SHAP zero enriching the frameshift frequency is the repetition of the GC motif at sites 17-18 and 21-22. This feature captures the microhomology-non mediiaa deletion of four nucleotides at 17-20, which results in one-base pair frameshift. Another top feature identified by SHAP zero depleting the frameshift fiequency is the repetition of the TC motif at sites 18-19 and 21-22. This feature captures the microhomology-mede meddateed deletion of three nucleotides at 18-20, which results in frameshift. These examples highlight the unique capability of SHAP zero extracting biologically meaningfully high-order interactions, which previously inaccessible due the space of feature interactions.
Capturing Epistatic Interactions in Protein Language Models. Protein language models have been shown to implicitly encode high-order (otherwise known as epistatic) interactions between amino acid sites [32 35 (5). This motivates the question: can we explicitly extract these interactions using SHAP zero? To explore this, we analyze Tranception K5 a protein language model trained to predict fitness from an amino acid sequence. Specifically, we focus on the green fluorescent protein (GFP), which has been shown to exhibit high-order interactions related its function [49]
51]. We
focus on evaluating predictions from Tranception over n = 10 epistatic positions from 49 (Fig.
Similar to the genomics experiments, SHAP zero's 's recovered Fourier coefficients (p 0.55) out-
performed linear (p= 0.48) and pairwise (p = 0.33) models in predicting protein function, further
reinforcing its ability to capture interactions (Fig. 4 4b). SHAP zero and KemelSHAP also showed
strong agreement in their estimates (p = 0.97) (Fig. 4b). Despite 4 being on the order of ~
1013
SHAP zero was still 3x faster in amortized time (5.2 vs. 17.7 minutes, respectively) (Fig. 4d).
Comparing SHAP zero to SHAP-IQ, the overlapping interactions correlated with p = 0.61 (Fig. 4e)
while running 7x faster in amortized time (5.2 vs. 38.1 minutes, respectively) (Fig. 4

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We analyzed SHAP zero's 's values (see Appendix FD and found that amino acids known to promote sheet stability, such as isoleucine (I) and valine (V), and alpha helix stability, such as leucine (L) and methionine (M), were strongly associated with increased fitness [52 Notably, proline (P) stood
out as a dominant amino acid associated with reduced fitness, aligning with its well-established However, SHAP zero's s interaction values revealed a more complex picture: despite its overall negative effect, proline can participate in positive epistatic interactions (e.g. when position 5 interacts with positions 7 and 9) when paired with specific residues like lysine (K) or asparagine (N), suggesting its disruptive influence can be contextually mitigated. Lysine also displayed strong context dependence, appearing in both favorable and unfavorable interactions. These results highlight SHAP zero's 's ability to uncover epistatic interactions at scale, feat previously unattainable. Details about all experimental setups can be found in Appendix
role as a secondary structure breaker due to its rigid backbone [53] 54
5 Discussion
Contrasting Different Amortization Strategies. From the amortization perspective, SHAP zero is closely related to SHAP estimation algorithms that use surrogate models to speed up computations [55]- 57]. One such algorthim, FastSHAP (55), trains a surrogate model to predict model outputs given a partially masked input. Although FastSHAP is theoretically optimal when the surtogate achieves the global minimizer [58], our results show poor empirical correlation with KemelSHAP on TIGER (p -0.39), inDelph (p = 0.08), and Tranception (o = 0.03) (Appendix F.. We attribute this failure to the nature of sequence models, where model outputs are highly sensitive to specífic masked features (e.g., seed regions in TIGER, cut sites in inDelphi, and epistatic positions in Tranception). Training a reliable surrogate would require model evaluations over nearly all unmasked combinations, which is infeasible. This contrasts with data modalities such as images, where the model output is rarely a function of few masked pixels. SHAP zero, on the other hand, solely relies on the assumption that the mode is compressible, leading an accurate and less biased amortization strategy with comparable speedup. Other methods that offer amortization assume that the input is binary (q =)=), 2), which are unsuitable for sequence models [59]]
Explainability Trade-offs in White-box Methods. White-box explainability methods, such as DeepSHAP [iI] and DeepLIFT [19], leverage the model architecture to estimate SHAP values extremely efficiently; however, they naturally fall short in (i) estimating high-order feature interactions, and (ii) black-box settings with only query access, which is the focus of this work. Regardless, SHAP zero due to its amortized nature, outperforms these methods in computational cost asymptotically. Extrapolating the amortized runtimes, our results in TIGER demonstrate that SHAP zero is more efficient when the queries exceed 25,000 sequences (see Appendix (F).
Limitations. SHAP zero provides accurate and amortized Shapley estimates under the assumption that the Fourier transform is sparse. However, this does not hinder the generalizability of SHAP zero. In sequence models, sparsity is more than just a theoretical assumption; it emerges from models learning the biophysical properties underlying Fourier transform reflects the presence of only a select few higher-order, epistatic interactions [60], a phenomenon consistent with biological studies [39, 51 61. Nevertheless, we emphasize that SHAP zero can adapt to varying levels of sparsity by tuning the number of samples while still amortizing explanation costs. For sparser models, SHAP zero requires fewer samples; for denser models, SHAP zero requires more samples. We can always take more samples to combat lack of sparsity. Our empirical results confirm this: SHAP zero took 4 million samples for inDelphi, 6 million samples for TIGER, and 800,000 samples for Tranception, but successfully amortized explanation costs across all settings. We provide detailed documentation on how to tune SHAP zero for future sequence models in Appendix G.
sepuence-function relationships. Sparsity in
Additionally, SHAP zero's strategy for marginalizing absent features is closest to the uniform approach proposed in [62]. Estimating the top-s Fourier coefficients by subsampling the sequence space is made under the key assumption that all q alphabets are equally probable. This contrasts with other Shapley-based algorithms [1] 25 26
28 62] which marginalize absent features based on the
empirical training distribution. For instance, in Tranception, the empirical distribution of amino acids
'Code and implementation for conducting the experiments can be found at https://gitthn com/ seairgroppccooee//baaazzooo Refer to the Appendix for more details.

---

in real-world protein sequences used for KemelSSHA heavily favors a small subset of amino acids (less than 5). biasing KemelSHAP to produce more negative SHAP estimates when encountering rare amino acid substitutions (see Appendix FF.. While SHAP zero and KerneISHAP estimates remain well correlated, they approximate slightly different quantities due to their marginalization assumptions. An interesting extension would be to marginalize absent features using the training data distribution, which would necessitate the development of sparse Fourier algorithms capable of subsampling the sequence space while accounting for this distribution.
Despite these differences, SHAP zero provides sufficient explanations identify both impactful amino acids and epistatic interactions. Comparing the top SHAP values from SHAP zero and KemelSHAP (Tables and (9. SHAP zero recovers all of the top positive amino acids and 17 of the 20 top negative amino acids identified by KernelSHAP. with the three missed negatives appearing in the bottom five of KernelSHAP's ranking. Thus, although SHAP zero tends to assign negative contributions more conservatively, it still captures the most impactful amino acids. To further assess SHAP zero's reliability, we examined the top five positive and negative interactions from Table(T) and verified that these interactions are epistatic in Tranception (see Appendix [.] for details). Taken together, these results demonstrate that SHAP zero reliably uncovers impactful amino acids and combinatorial interactions at scale.
6 Conclusion
In this paper, we introduced SHAP zero, algorithm for estimating Shapley values and interactions with a near-zero additional cost for future queried sequences after paying an initial up-front cost to find the Fourier transform. Our theoretical analysis and large-scale experiments show that SHAP zero significantly reduces amortized computational cost by orders of magnitude while identifying nearly all predictive motifs. Overall, SHAP zero expands our toolkit in the explainability of biological sequence models and, more broadly, in problems with a combinatorial nature. Our work will further encourage interdisciplinary algorithms and theoretical frameworks at the intersection of signal processing. coding theory, and algebraic geometry for the benefit of explainability in machine learning.
7
Acknowledgment
This research was supported by the National Science Foundation (NSF) Graduate Research Fel- lowship Program (GRFP), the Parker H. Petit Institute for Bioengineering and Biosciences (IBB) interdisciplinary seed grant, the Institute of Matter and Systems (IMS) Exponential Electronics seed grant, and Georgia Institute of Technology start-up funds.
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A Computing Shapley Values in q-ary Functions
The exact computation of Shapley values requires querying f over an exponential number of query sequences. To illustrate this, we will walk through an example of computing the Shapley value I x; (j = 1) for a q-ary function that takes as an input of alphabet size q and sequence length of 3. For brevity, we will drop the notation x,.
For simplicity, let assume the following function for ʃ and are trying explain the sequence x 1,,1,0
((m) m_ mg + mg:
where m, indicates the tt coordinate of m. In this example, it follows from n 3 that D 21)
The total possible subsets T S DV = 1} can then be written out as: [0. (3), (2, where is the empty set. Expanding out the Shapley value equation from the Introduction, we then have the following:
7 (j = 1) 01(3-0
1)1
[x)1)(x) (x))
3!
11(3
1)
(1112)) (x) ((2) (x)]
3!
1!(3
1)!
[U(13]] (x) (xa]
3!
21(3
2
1)!
[112233) (x) [x)]
3!
[1)](()) - (x)) + 1 (1)) - ((2
(x)))()) (x)] (1))) 112.33 (x)].
From here, we compute each value function, which requires fixing features in T and marginalizing the rest.
nx(x)=
23 23 (5(1000,)) + ((1,0,,0)) + ((0,1,0) + f(00,0, 1)) + ((1,1,0)
f() + ( 3 (x) =
2
2 (f(10,0,0) ((0000)))) f(000 0, 1)) + f((0, 1, 1,)))) 1.
((2) =
2 [f((0, + f(1,,,,)))) (.) + f([1,1,1)) =.
((3) (x) 2 (f(10,0,0) + f([1,0,0 f(00 + f(L,1,0) 1.
3
1(1.2) (x) 1,0) + f((0 1, 1])
2
1(1,3) (x) 2 (f((0,0,0) + f((o, 1,0)) = 2
((2,3) (x) =
2 (f((0, 1,0) + f(1111, 1,0) 2
(x) 20 (f(10, 1,))) = 1.
Due the reliance of q in the value function, Substituting all of these value functions into our Shapley value equation:
required to loop over all q" sequences.
3
1.3
3
sS ys 1)
2
1
2
2
6
2
1
1
1
1
1
6
12
12
6
2
15

---

B The SHAP zero Algorithm
Algorithm SHAP zero: Estimating Fourier Coefficients (One-time Cost)
Require: Alphabet size Subsampling dimension b, Number of subsampling groups C, Number of offsets P, Subsampling matrices (Mc)ee[C)- Offsets (E])
1: F с-
2: [Initialize recovered Fourier coefficients)
3: for each [C] do
4: for E [P] do
5: for cach E 2g do
6: Ueppli] qt EeERY S(M + w..)
7: end for
8: end for
9: end for
10: Build sets S ((c,j) U.. [i] is singleton), and L {(c,)) Unp [j] is multi-ton)
(Peeling)
12: while [S] > 0 do
13: for each [c E S do
14: Recover k and F[k] from [ip
15:
F F-FU, F(k)
16:
for each c' E [C] do
17:
j'+ --M
18:
Ue, + pD1 (d2
end for
20: Update S, L accordingly
end for
22:
end while
return F (Fourier transform estimate)
Algorithm 2 SHAP zero: Computing SHAP Values and Faith-Shap Interactions (Per Query Sequnce)
Require: Input query sequences {x, heto) Fourier transform estimate with maximum order \
I: for each input query sequence x; do
2: (Compute Möbius transform around x;
3: for each k 2 do
4: Mx, [k]
Eyezy Fly]wixiss (-)lk(- wm
5:
end for
6:
(Map Möbius transform SHAP values)
7:
for each feature j = 1,..., n do Eks, [K]log Rolo Mx, [k]
8:
1SV (j)
9:
end for
10:
Φi [TSV (1), (1),-... TSV (n))
11:
[Map Möbius transform to Faith-Shap interactions)
for each interaction set T E Tr do
13:
FSI (T) = (-1) Lkkrr0m Gाl Mx, [k] 14: end for
15: T; + FFSI [T)
VT ET
16: end for
17: return (Φ. 2 hiel))
B.1 Shapley Explanations, Möbius Transform, and Fourier Transform in SHAP zero
The connection between Shapley explanations, the Möbius transform, and the sparse Fourier transform lies in how we represent and recover feature interactions in complex models. SHAP values attribute
16

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model's prediction to individual features and their combinations by decomposing the function into additive contributions over subsets. Simiarly, Faith-Shar interactions extend SHAP values over interactions of features. These decompositions align with the Möbius transform, a mathematical framework that captures how a function defined over feature subsets can be expressed in terms of unique contributions from each subset. However, in practice, the exact computation of this decomposition is infeasible for high- high-dimensiooal problems due to its exponential computational complexity. The sparse Fourier transform offers computational shortcut by exploiting the empirical sparsity of interactions in biological sequence models.
The only approximation in SHAP zero is computing the sparse Fourier transform. Given an exact Fourier transform and infinite compute power, one could compute the Möbius transform and, conse- quently, SHAP values and Faith-Shap interactions, exactly. Thus, the quality of Shapley estimates is heavily dependent on how well the Fourier coefficients approximate the model. To facilitate the easy usage of SHAP zero, we have released a pip package that automatically tunes the parameters of SHAP zero and provides the best estimate of the Fourier transform given user's computational budget.
C Computing the Fourier Transform
C.1 Example of Sparse Fourier Tranform
The key to SHAP zero lies in the assumption that the model is s-sparse in the Fourier domain. To illustrate an example of sparsity in Fourier, consider the following function, defined over n 3 sites for q = 4 alphabet, modeled using Equation
(3):
f(m) = (5 + 2jjw (m.,00.1,0) (3 + j)w (m.20.000)) + (1 j)w (m.,
ff contains the Fourier coefficients F[ 1, 0] = + 2j, FI11, 0]] = - (3 + j). and FI 0, 3] = 1 1+ j. The Fourier transform is sparse with = 3 non-zero coefficients out of q" 4 = 64 possible coefficients. Additionally, the Fourier transform also has a maximum interaction order of = [[ 0, 3|lo = 2).
(l][1,2, 0]Il-
C.2 Estimating the Sparse Fourier Transform
In this section, we add more details about finding the sparse Fourier transform. The procedure SHAP zero takes can be broken down into three steps: 1) Subsampling and aliasing, 2) Bin detection, and 3)
Peeling.
Subsampling and aliasing, From Equation (4 we note that the coefficients Ue. are aliased
versions of the coefficients F []. We denote J. as the grouping of subsampled Fourier coefficients
U. = [Uc.ii]..... Ue. [i]]. with their respective offsets also grouped up into the matrix D e
Pxn We rewrite Equation as:
U.
Σ F.k.
(11)
k: MT
We additionally denote Se,k = w D.k which allows us to re-write the above equation as:
U.li]
Σ
Flk]sa.k
(12)
M-
Noiseless Bin Detection. After aliasing f. we aim to recover the Fourier coefficients using bin detection procedure, where we aim to identify which U.(i) contains only one Fourier coefficient. For simplicity, we break this into two sections: noiseless (when the Fourier transform is perfectly s-sparse) and noisy (when the Fourier transform approximately s-sparse) bin detection.
In the noiseless case, suppose that for each subsampling group c, we choose M. to be partial-identity matrix structure of the form:
M. [0xxt(e-1) 1); Lxbb, (nx
(13)
Fig- D shows an example of the aliasing pattern with the partial-identity matrix structure. Additionally, choose P (the number of offsets) and D. Lxxn. We also choose fixed delay de O..
17

---

where On is the vector of all O's of length n. Consequently, Uc.o = EKMTK= Fik). We then use Ue,o[j] to identify U.[]] as a zero-ton, singleton, multi-ton bin based on the following criteria:
Zero-ton verification: U.[]] is zero-ton if Fik] for vectors k such that M (the summation condition in Equation (12 In the noiseless case, this is true when Ue Ji = 0 for all
p = .........
Singleton verification: U. is singleton if there exists only one k where Fk] + 0, given M = J. In the noiseless case, this implies that [V,, = (V,,2 = = cU [i]l- This
allows us denote Ue] as a singleton if the following condition met:
UeP [i]
=1, p=
Multi-ton verification: Ue]i] is multi-ton if there exists more than one where F(k) * given M = In the noiseless case, this means that:
Uepp Ue.ojil
# 1, p
P.
If a singleton is found, we know that U.. = F[k]. The only unknown is what the vector k is. By exploiting the identity structure of D., we know must satisfy:
arg, [U,, [U]
=D.k.
(14)
agg, (V,,lu,,lii/// c.,
Here, arg, - Bn is the q-quantization of the argument of complex number:
arg (z) =
arg ze
(15)
2m
Therefore, after obtaining the vector k from the above equation, we know Fik] U.,o
Noisy Bin Detection. In the noisy case, we cannot assume that most coefficients are exactly zero. We model this problem as one where the subsampled Fourier coefficients are corrupted by noise:
Ucplill =
Σ
F[k] (dop.) + We,
(16)
k: MA
where Wa [i is Gaussian noise with zero mean and variance v², such that 2² = 0² /²/B.
To recover the sparse Fourier transform, relax the constraints made M. and D in the noiseless case. We let M. be random matrix in Xn and given the hyperparameter Pl, we let P = P n and D. be random matrix in Pxn, In the presence of noise o², which is hyperparameter in practice, for some (0, 1), we do bin detection using the following criteria:
Zero-ton verification: U.]i] a zero-ton U.DIi < (1+7)1².
Singleton verification: After ruling out zero-tons, obtain an estimate k, which we denote as k. Then, estimate F] as S P. We can then check [ is a singleton if:
- [[ - F[k]s < 71+?)
Multi-ton verification: If U.j neither a zero-ton nor a singleton, then U.(] is multi-ton.
SHAP zero obtains an estimate of k for singleton detection based repetition coding. Given our P1 random offsets d such that P = Pin, we construct perturbed versions of each offset by modulating with each column of the identity matrix such that we have offsets dp.r:
dp. dp 0g er, Vp [Pt), E
(17)
18

---

This allows to employ a majority test to estimate the pth value of k:
ki = arg max
Σ
1{a= arg /upp))
(18)
aE
pE
where 1 is the indicator function, SHAP zero uses noisy bin detection to recover the sparse Fourier the sparse Fourier transform in practice can be found in
transform. Details about how Appendix G.I
Peeling. After determining the bin type, we apply peeling decoder to recover additional singletons. The problem is modeled as sparse bipartite graph: each F k] corresponds to a variable node, each U.li] is check node, and edge connects F[k] and U.] when M j..
Upon identifying a singleton, for every subsampling group c, subtract F[k] from the check node U.[Ma This effectively peels an edge off the bipartite graph. We repeat this process over all subsampled Fourier coefficients, iteratively peeling singletons until no further singletons remain.
After peeling can no longer be done, SHAP zero recovers all Fourier coefficients with a sample complexity of Osn?²) and a computational complexity of O(sn)). We refer readers to 40 for comprehensive theoretical analysis.
C.3
Sparsity and Interaction Order in Encoding q-ary Functions
Our theoretical analysis shows that the performance of SHAP zero depends on and f. A natural question arises: does the choice of encoding scheme in g-ary functions (c.g., in DNA, A = 0, C = = 2, = versus C
= 1, G 2, = 3) affect ??
We emphasize that SHAP zero is robust to the choice encoding scheme. Both s and & are intrinsic to the underlying sequence- funetiio relationship. By the shifting property of the Fourier transform (Equation (21) in Appendix I..I.. changing the encoding scheme will only change the magnitude of non-zero Fourier coefficients, leaving and untouched. Consequently, the theoretical guarantees of SHAP zero are unaffected by the encoding scheme.
To validate this empirically, consider multilayer perceptron trained RNA sequences of length n = from (63 predicting splicing efficiency [64]. We ran SHAP zero under four encoding schemes (see Table 1 and, following the methodology described in Appendix G.I. computed the R when reconstructing model predictions on 10, 000 random sequences using the recovered Fourier coefficients. Although varied across encoding schemes (437 + 119), likely due the model not being exactly sparse, R² values remained highly consistent (0.78 0.01), and & was consistently 4. These results empirically demonstrate that SHAP zero is robust the choice of encoding scheme in real-world models.
Encoding scheme
R2
A = 0
C ==1
U = 2
G == 567
0.80
4
C
2
U
G 326
0.77
= 2
C 3
U =
G == 345
0.78
4
A =
C C=0
= G=2 508
0.77
Table 8, R2 and across different RNA base encodings when recovering Fourier coefficients with SHAP zero.
D Computing the Möbius Transform
D.1
Set Möbius Transform
each set Möbius transform coefficient alv, S), where S C T, represents the marginal contribution the subset S, given a value function v, in determining the score (x(x). Reconstructing the original score (7(x) from all subsets S C requires taking linear sum over all the marginal contributions. For instance, in an n = problem, say we want to compute the set Möbius transform of the set S {1, 4). Let ve (x) be the output of the value function when
From Definition 2.1
19

---

inputs are given. The set Möbius transform a(u, S) can be written as:
a(v, (1, 4)) = (-
+ (-
+ (-1))((
(-1) (- cx))
Each exponent raised to the negative one corresponds the sign of each value function's output. Simplifying each exponent results in the following:
a(e, (1,4)) (x) - "(1)(x) - (x))(x) + vx()).
compute the inverse Möhius transform of the set T {1,4). The value function can be written in terms of a(u, S):
Now, suppose
(x) = a(v, 0) + ale, (1)) + a(v, (44) + a(v, (1,4})
D.2 g-ary Möbius Transform
Examples of Standard Real Field Operations. Computing the q-ary Möbius transform into the g-ary domain involves utilizing standard real field operations over 2 + R. To illustrate the subtraction of m, consider the vectors [1, 0, 0, 2] and m = [1, 0, 0, 0j. Then,
k o = ||
[1,0, o, 0]1lo = || 0, 0,0,,]llo= 1
This is only computable when m 5 k. For example, the vectors m 01,0,0.0) and k = [1, 0, satisfy S k: mi Kj 1, m2 h2 0. mg h3 and m4 0. An example of vectors that do not satisfy m < k are m [1, 0, 1] and k 0, Here, my + kis 0 and m4 = 1 K1
so m is violated.
Computing the q-ary Möbius Transform. To illustrate computing the q-ary Möbius transform from f. consider function where q 4 and n and define the encoding scheme as A = 0, C T= 2, and G = 3. Suppose the query sequence is x; = [2. 1, 0,3) (corresponding to the sequence "TCAG'). We wish compute the q-ary Möbius transform Mx, [k] for k 0,,0, 0,
Using the definition from Equation (5.
we have:
Mx, [1,0,
Σ
(-1)1l[1..00.))-m)) ((m + x;) mod 4).
(19)
m [1,00.
Expanding the sum over all m & [1,0,2,0}:
Mx. [[, =(=)))((11,000022221133 mod 4)
+(1)))(((.00000)+221103 mod 4)
+(=1))((0000022))22110 mod 4)
+ +(=0)3((0.00000)2221... mod 4).
Simplifying each term of results in:
(1.100..) [3, 1, 0, 5) 1,0,5 mod 4 [3, 1, +1.00,,0 = [3,1,033]: 1,0,3 mod 4 = [3, 1,
[0.0.0.2] [2,1,0,5]; 1,, mod 4 [,, 1, [0,0,0,0] 3, 3,,3; [2,
mod 4 = [,,1,0,3]
This allows us to simplify as follows:
Mx, 0, 0, 0, 2]] = f(33, O, 1]) - f(3, (,0,3) - 1,2, 1, +12,2,,0,3)
Computing the Inverse q-ary Möbius Transform. Now, we illustrate computing the inverse q-ary
Möbius transform to recover of from Mx.- Suppose again the same query sequence [2, 1, except we wish to compute f(mm + x;) mod 4) for m 0,0,0,2
Using the definition from Equation (6). we have:
f ((11, 0, + x,) mod 4)
Σ
Mx. [k].
(20)
20

---

Expanding the sum over all k E 0,,0,
f ((1L, 0, 0, + (2,1,0,3) mod 4) Mx, [0000,,0 + Mx 0,
+Mx, [00,0,0, 2]] Mx. |[, 2|].
This results in the following:
Mx [00, 0,0, Mx. [1, 0I] + Mx [[0, 0. 0.0. + Mx [[1, 0, 0.]. f (3, 1,0, 1))
E
Proofs
In this section, we provide proof' for the Proposition 3.2 Proposition 3.3
Theorem 3.4 and Theo-
remb 3.5 For simplicity in notation, we drop the subscript from x; and write x.
E.1
Proof of Proposition 3.2
In the Fourier transform, all alphabets in q are treated dictionary that maps an input to g-ary value. However, since the q-ary Möbius transform generalizes the S operation in Z R, for each input query sequence x, we utilize the shifting property of the Fourier transform. The shifting property of the Fourier transform states that every Fourier coefficient must be shifted by the sample's q-ary encoding using the following equation [40]:
Fly] Fiy] Vy Z
(21)
where Fly] refers to the shifted Fourier coefficient, x refers to the Fourier-encoded query sequence, and Fly] refers to the original q-ary Fourier coefficient. To shift the entire s-sparse Fourier transform, we have to loop over all Fourier coefficients and shift them by the query sequence.
As consequence of the shifting property of the Fourier transform, can re-write Equation as:
f((m+ + x) mod q) = Σ
(my)
(22)
yEΣ,
To illustrate this further, consider the query sequence [0, 1, 2). assuming the encoding pattern where A. C, 2 G. and T. The shifting property of the Fourier transform allows us to characterize this same sequence as [0, 0, 0J, which satisfies the < operation (see Appendix D.2).
Generalizing this, we aim to shift the Fourier transform in such way that our sequence x is encoded as O,,-
After obtaining [ly], we plug in Equation (2 into Equation 5 to obtain:
M
(-1) Ik--ll
Σ
(m)
(23)
mskk
y
By pulling Eyezy Fly] out of the summation Emsk( q-ary Fourier transform to Mx(k] as given in Equation
(-1) I]lk millk-mm mllo, obtain the mapping from the
Mx = Σ Piyl Σ
(-1)ilk m (lljikk-mloowmmms)
yEZ
mu
= Σ (xy)
m-ll msk
Vk E z
yE
Equation 77) details how to convert the shifted Fourier transform Mx(k] for a single single q-ary Möbius coefficient). If no assumptions are made, the complexity of mapping the Fourier transform into every possible Mx kk scales with O(a*). However, we assume the Fourier transform is low-order and has maximum order of f, we can bound the number of nonzero elements in M [k]. This is because Equation 7 will always go to zero when there exists an such that y; = and k; 3 As example, consider finding the q-ary Möbius transform of function which is 1-sparse
21

---

We now move to analyze the computational complexity of converting from the Fourier to the q-ary Möbius transform. First, shifting all the Fourier coefficients scales with Oss), since we have to loop through all Fourier coefficients once. The time complexity of doing the conversion using in Fourier and let assume F[[2, 1, 0]] is the only non-zero Fourier coefficient. In this setup, one example for which M2 [k] is zero k [0, 1, since it satisfies the condition that 3/3 0 and k3 * O. Computing Equation 7 we get:
M₂ k) F1 1, oi]
(-1)ll0. -00.--00,0000 10000000000000000000000000000000 + (-1) 10.1. -0.1111100...001111 of (1.
+ (-1))l10.11)) 100000000111100_00000111111 + + (0,11, (0.00,0))0. )
From here, simplify to the following:
Mx F[[2, 1. (-1)2w² (-1)'" +-11)'0 (-1)
As a consequence, now we can bound the number of nonzero elements in MgM [k] by counting the number of vectors k for which there is i such that ys 0 and k; * 0. If we assume that the Fourier transform is low-order and bounded by É, then we can bound it by (44") instead of (gg)).
Generalizing this example, if there exists an i such that = 0 and k; * 0, this allows us to rewrite Equation 7 as the following:
Mx =
Σ y yE
Flyl
Σ C-11) (-l) + (-1)lkk (-1)lik-m "lo 'loo- y)
msk
m0
where m+ is equal to m everywhere except at location and m = k; Since = O, (m, (m+ y). Additionally, we can rewrite || 1-mmlo m|lo as || m=
flo + 1, since know that the only difference between m and m is a single nonzero value at position i. Therefore, we can say that:
M₂ [k|
Σ Flyl yE
Σ mmsk
(-1) (ml (my)) + (-11) [l]lk/-mllp ml mlotl (( 0.
Equation depends the distribution of the Fourier coefficient interaction orders. In the worst-case scenario, we can assume that all Fourier coefficients are all ε-th order. We break up the problem into two categories: the computational complexity of computing the most expensive q-ary Möbius coefficient and analyzing the total amount of q-ary Möbius coefficients created. From Equation (7). the most expensive q-ary Möbius coefficient to compute will be l-th order Möbius coefficient due to summing over 2' vectors, which is the total amount of vectors encompassed in m < k; this single q-ary Möbius coefficient requires summations over 2 vectors, which scales with computational complexity O(s2t), ). Additionally, if all Fourier coefficients are ê-th order, there are at most sqe possible q-ary Möbius coefficients loop through. Therefore, the overall computational complexity of converting from the Fourier to the g-ary Möbius transform is O(s + s2 x sqe Rs (2)
E.2 Proof of Proposition
3.3
To prove Proposition
3.3 we require using the q-ary value function, which we leave here for reference:
Ur (x
Σ
f(m).
(24)
qiT
mi
We plug in the inverse g-ary Möbius transform from Equation 6 into the q-ary value function. By using Mx [x[k), this means that we shift m such that mr = O Here, value 0 is assigned to subset indices T", which indexes the subset of the original inputs xy. Over the missing inputs x and their
22

---

respective indices T, the vector my E 2gr is used, by definition of the value function:
oT(x)
Σ
f(m)
1
Σ m mr=0r
f((m x) mod g)
qiT
Gi
mimpER mT=X7
Σ E2
Σ ASm
Mxlk)
q
12
Due to the recursive nature of the q-ary Möbius transform, each q-ary Möbius coefficient is looped through qf I-IIllll times. Therefore, simplify follows:
UT(x)
Σ Em
| lmmll
M Mx[m]]
my=0
We can pull the q! out of the summation, which will allow us to cancel the y q term in the denominator. This leaves us with the following:
(T(x)=
Σ m:my Ez(21
qlimila Mx|m]
(25)
my
Plugging Equation (25) into the forward set Möbius transform equation from Definition 2. returns an equation for the set Möbius transform in terms of the q-ary Möbius transform:
() = Σ (-1))S-ITT
Σ
Mx [m]
(26)
qllmilo
TCS
mm my=0
Although Equation (66 loops only over subsets T S S, all possible q-ary Möbius coefficients accounted for, since m is nonzero at coordinates given by T. We rewrite Equation (26) such that we loop over all possible q-ary Möbius coefficients, given by the subsets L C D:
Ma(ma] )
alv, 9)
Σ LCD
Σ T-ICS LLTT
1S--TT
Σ
M. m2=0 m
Notably, are looping through T S over the set T in order to compute (-1)151-1T), but our if condition is with respect to T. For every T, T is implicitly defined already, since the complement T.
We look to simplify the outer summation. Consider the relationship between an arbitrary given the condition L C T. L C T' is only true when L n T = 0 based on the way T is implicitly defined. Since T C S, the total number of features that can be in T that can satisfy L n T is |s L). Therefore, the amount of subsets that will loop through T C S is equal to 2/8 for an arbitrary L. From here, will cancel terms by noting that must loop through the alternating series 218\2/ times. We divide into two cases: when S C L and when S I L. When S & L, I.S L > which means we loop through the alternating series an even amount of times, canceling out any g-ary Therefore,
Möbius coefficients that have [T] nonzero values. However, when C [. we can rewrite the outer summation to be over all possible L's where S L:
alv,S)=
Σ
Σ
(-1)181-17)
(
Σ miViEL, mz=0
(//m))
L.SCL
T-TES it LnT==
Now, given C T., the only way C S and I. 7 can be satisfied when 7 0. cannot contain any features in its set because it will violate L n. T 0. To illustrate this, consider
23

---

problem. Let S = {1. 2} and L = (1.2.3}. The possible subsets 7 C 8 include 0, {1}, {2), and (1. For the subsets {1} and {2), (1,
[1] + and {2) n {1, 21,2,3 {2) * 0. The
only case where 7. n 7 = 0 is when T = 0, which means [7] 0.
Setting [7] 0 and the outer summation to loop all possible I/s where S C T., obtain the following expression:
S)
Σ
(-1)1s)
Σ
Malml]
L.SCL
m.
m.
Pulling (-1) out of the summation, we obtain an expression to convert from the g-ary Möbius transform to set Möbius transform:
au S) = (-1)18)
Σ
Σ
Mx|ml
(27)
q
L.SCL
m.ViLL m2=00/a
This can equivalently be written in vector notation
a(u, S) = (-1)1s)
Σ
Mx [k].
(28)
kllo
k:k
0181
E.3 Proof of Theorem 3.4
With conversion from the q-ary Möbius transform to the set Möbius transform, now derive the SHAP value formula with respect to the q-ary Möbius transform. To do this, we will need to use the following Lemmas.
Lemma E.1. For all n 2 0 and 2 O, where n and are integers:
n k n
l:
k
Proof. Expanding the right-hand side results in the following: (x=1) (n-1))
(
k-n
(A (1 EJT (1=1)-
Lemma F.2. For all n 2 where n is an integer:
Σ
11
1.
Proof. We require special case of the binomial theorem. The binomial theorem states that, for any
number and y, and given an integer n 0: (x y)" Σx (#)="y" Plugging in
and æ = -1 leads to the following simplification: Σ" O-10 =0 We pull out the summation
term where k = 0, and instead loop the summation over from A = 1 to n: 1 1- 0.
Finally, isolating the summation term leaves with Lemma E.2 (-)(
Our derivation will require the classic SHAP value equation, which we leave below. Since we drop the i subscript from X;, we write 18V (i) instead of x; (j):
T T 1)!
s Σ
(x) uT
(29)
D
TEDY()
To begin our derivation, we plug in Equation
27 into Equation
29
Mx [m] ))
SV ii)
Σ TCDUO
(-I)
Σ
Σ
(30)
TU
GIEl
LTUICCL
m. >0
m, 15.=0,p

---

Summing up over all sets T S D {i} results in repeatedly looping over the same q-ary Möbius coefficients several times. We re-write Equation (30) sum over all q-ary Möbius coefficient sets S D (i), which allows us to loop over all unique q-ary Möhius coefficients as we keep track of how many times the alternating series gets summed:
(1)
(1)
E LED
1
Σ m my
Mx
Σ
(-)))T))
(31)
GILUEEH
TELU(i) iET
m -0
The amount of times to sum through ETcLu(),ie) [T7 (-1) depends on 77. For example, consider an 3 problem, where
and = (2,3). There only one possible where | = 1. {1}, because i T. However, there are two possible sets where TT' = 2: {1. 2} and {1, 3). Generalizing this, the amount of possible sets for an arbitrary [T'] that satisfy
is equivalent to
[ (LLu((H-)). Using this, we re-write Equation 31 by summing over all possibilities of [T]:
LUU()) Σ
[U {i} - 1 )) (EU k-1
V Σ
Σ
Mx(m [m]
IEDVO
mXjjELu() m,,0 mp\(s] AA(G)
We can apply
Lemmas E. and
arrive at the following:
(LLU (-1)+)
S (i)
Σ GiLUEHH
E
Mx(m)
LU [i]
LCDWE)
mi m, mA40) = (are
k=1
(1) = E
1
Σ (i))
Mx|m]
L
{
LCDWA
mpx(a 1xx001
Finally, we can pull [Zu[s out of the summation be left with our final expression:
TSV (1) =- Σ
Σ
Mx[m].
(32)
LCDWA
This is equivalently written in vector notation as:
(V(())
(1)
Σ k:k;00
|kl Mx [k..
(33)
Analyzing the time complexity, in the worst-case scenario, if every q-ary Möbius coefficient is nonzero up order É, the total amount of coefficients to sum up is q² (q 1) q². Therefore, the time complexity of Equation (33 is upper-bounded by Oga²).
E.4 Proof of Theorem 3.5
The Faith-Shap equation given in Equation 2 valid under the assumption that both f and I IFSI (T) constrained to a order f. If f is not constrained by order t, we compute I FS/ (T) using the following equation (26):
T 8 T
(1)
1 - +
alv, S),
VT E Tr (34)
(T)
S|--t-1 E+1T
SSY. S
where T₃ denotes the set of all T S D where [T) < l.
This formulation is not computationally scalable, the Möbius transform requires looping 2ID) subsets. However, for g-ary Möbius transform with maximum order e, all set Möbius
25

---

coefficients with maximum order greater than are 0, by Equation (27). Therefore, when computing the hd order -itth-Shap interaction index, there are no set Möbius coefficients that satisfy |S| in the summation term of Equation (34), allowing us to drop the summation term. Therefore, we can define the ehh order Faith-Shap interaction index equation as follows:
FST (T) = a(v,T), AA Tτ.
(35)
From Proposition 3.3 this is equivalently in vector notation as:
FF (T) = (-1)
Σ kmr>00
glikllo Mx(k).
(36)
with computational complexity of Ogff).
Similarly to the computational complexity from Theorem 3.4 every q-ary Möbius coefficient is
nonzero up order f, the total amount of coefficients sum up - (q 1) z q'. Therefore,
the time complexity of Theorem 3.5 is (g))) Notably, when e = 1, Theorem 3.5 becomes identical
to computing SHAP values in Theorem 3.4
26

---

Additional Experimental Results
Top positive SHAP values
Top negative SHAP values
Position
Nucleotide
Average SHAP value
Position
Nucleotide
Average SHAP value
5
0.120
-0.079
6
0.118
I0
A
-0.076
7
G
0,087
6
-0.073
25
0.080
A
-0.071
22
T
0,070
9
A
-0.069
C
11
0.061
20
-0.064
21
T
0,061
5
-0.055
10
0.061
C
-0.050
8
G
0.060
6
T
-0.047
3
0.058
-0.045
9
G
0.053
21
C
-0.045
4
0.047
-0.043
8
C
0.043
22
-0.042
2
A
0.043
-0.041
T
23
0.042
25
-0.041
G
4
0.037
-0.041
C
9
0.036
4
-0.039
10
0.036
18
-0.037
24
0.032
11
A
-0.037
18
0.031
17
-0.037
Table 2: Top 20 positive and negative SHAP zero values in TIGER from SHAP zero when predicting the guide score of perfect match target sequences. The top SHAP values are centered around the seed region and are in agreement with KernelSHAP.
27

---

Top positive SHAP values
Top negative SHAP values
Position
Nucleotide
Average SHAP value
Position
Nucleotide
Average SHAP value
6
0.123
10
-0.066
5
G
0.119
7
A
-0.065
7
0.094
8
A
-0.062
PARN
11
0.076
-0.062
25
0.073
20
-0.060
C
4
0.070
-0.058
10
C
0,062
3
-0.046
8
0.059
-0.044
3
C
0.055
9
-0.041
8
0.054
-0.040
9
C
0.052
11
-0.040
0.050
-0.040
22
T
0.049
8
-0.038
0.049
-0.038
9
G
0.047
22
-0.037
4
0.046
23
-0.037
12
C
0.039
21
-0.037
18
A
0.038
10
-0.036
10
G
0.037
5
-0,036
23
0.036
-0.035
Table Top 20 positive and negative SHAP values TIGER from KernelSHAP when predicting the guide score of perfect match target sequences. Similar to SHAP zero, the top SHAP values are centered around the seed region.
28

---

Top positive interactions
Top negative interactions
Positions
Nucleotides
Average interaction
Positions
Nucleotides
Average interaction
5,6
G.T
0.061
3,4
C,C
-0.097
5,6
C.A
0.061
8,9
C,C
-0.079
3,4
A.C
0,056
3,4
G,G
-0,061
3,4
C,G
0,055
4,5
G,G
-0.059
19,20
C,A
0.052
4.5
C,C
-0.059
5,6
A.C
0,052
6,7
C,C
-0.059
5,6
T.
0.052
6,7
G,G
-0.059
19,20
A.C
0,046
5,6
G.G
-0.059
4,5
T.G
0,044
5,6
C.C
-0,059
4,5
A,C
0.044
9,10
T,T
-0.059
8,9
A,C
0.043
19,20
A,A
-0.059
18,19
C.A
0.041
10.11
G,G
-0.057
4,5
G,T
0.038
10,11
C,C
-0.057
4,5
C,A
0.038
9,10
C,C
-0.056
10,11
A,C
0.038
11,12
G,G
-0.556
10,11
TG
0.038
11,12
C,C
-0.056
8,9
C,G
0,038
8,9
G,G
-0.054
11,12
A.C
0.037
7.8
T,
-0.052
11.12
T,G
0,037
7,8
G,G
-0.052
6,7
T.G
0.037
7,8
C,C
-0.052
Table 4: Top 20 positive and negative Faith- Faith-Shap interactions in TIGER from SHAP zero when predicting the guide score of perfect match target sequences. The top interactions exhibit high GC content around the seed region.
29

---

Top positive SHAP values
Top negative SHAP values
Position
Nucleotide
Average SHAP value
Position
Nucleotide
Average SHAP value
19
3.727
19
-5,462
19
A
2.466
18
G
-3.105
20
2,431
20
-2.981
21
1.833
21
-2.891
22
1.754
18
A
-2.617
22
1.751
-2.539
18
T
1.734
17
C
-2.495
21
1.733
17
A
-2.495
17
G
1.696
17
T
-2.461
20
A
1.649
22
-2.451
18
1.527
23
-2.266
17
1.492
-2,239
21
1,490
22
G
-2.223
18
C
1.490
18
-2.138
T
21
1,422
22
T
-2.120
22
C
1.355
22
-2.034
C
23
1.220
20
T
-2.001
16
C
1.177
17
G
-1.972
22
A
1.121
20
G
-1.877
17
1.104
18
-1.775
Table 5: Top 20 positive and negative SHAP values in inDelphi from SHAP zero when predicting the frameshift frequency of DNA target sequences. Top SHAP values are centered around the cut site and are in agreement with KernelSHAP.
30

---

Top positive SHAP values
Top negative SHAP values
Position
Nucleotide
Average SHAP value
Position
Nucleotide
Average SHAP value
19
4.383
19
6.148
17
G
4.308
18
G
-4733
20
3,585
23
-3,483
18
2.773
17
C
-3.212
19
A
2.649
20
-2.650
20
2.497
-2.641
22
G
2.359
22
T
-2.534
21
T
2.253
A
-2.507
14
G
2.237
17
-2.458
A
2.161
19
C
-2.305
18
2.156
22
-2.282
18
A
2.117
20
G
-2,247
21
C
2.085
15
G
-2.235
17
2.013
G
-2.116
22
T
1.999
20
-2.164
23
1.921
18
A
-2.144
21
G
CTTI
1.838
18
-2.116
G
15
1.648
26
-2,091
14
1.580
15
T
-2.009
23
1.528
22
-1.959
Table Top 20 positive and negative SHAP values inDelphi from KernelSHAP when predicting the frameshift frequency of DNA target sequences. Like SHAP zero, the top SHAP values are centered around the cut site.
31

---

Top positive interactions
Top negative interactions
Positions
Nucleotides
Average interaction
Positions
Nucleotides
Average
interaction
18,19,21,22
G,C.G,G
2.504
18,19,21,22
T,C.T,C
-5.640
18,19,21,22
C,G,C,C
2.504
18,19,21,22
A.G.A.G
-5,640
18,19,21,22
A.G.A.C
2.504
18.19,21,22
G,C,G,C
-5.640
18,19,21,22
T.C.T,GG
2.504
18,19,21,22
C,G.C.G -5.640
18,21
G,C
2.073
18,19,21,22
G,G.G.G -5.400
18,21
C.G
2.073
18,21
G.G -.325
17,18,21,22
G,C.G.C
1.983
18.21
C.C -5325
17,18,21,22
G,G.G.G 1.983
17,18,20.21
G.A.G.A
4.658
17,18,21,22
A,A.A.A
1.983
17,18,20.21
T.G.T.G
4.658
17,18,21,22
A.G.A.G
1.983
17,18,20,21
A,T,A,T 4.658
17,18,21,22
A.C.A.C 1.983
17,18,20.21
C.A.C,A 4.658
17,18,21,22
T.T.T
1.983
17,18,20,21
T.C.T.C
4.658
17,18,21,22
T.A.T,A 1.933
17,18,20,21
G,G.G.G
4.658
18,19,20,21
C,T.C,T 1.982
17,18,20,21
C,G,C.G
4.658
18,19,20,21
A,G.A,G
1.982
17,18,20.21
G,C.G.C
-4,558
18,19,20,21
C.A.C,A 1.982
17.18,20.21
A,G.A.G
4..58
18,19,20,21
G,G.G,G
1.982
17,18,20.21
A.A.A.A
-4,588
18,19,20,21
G.A.G.A
1.982
17.18,20.21
T.T.T.
-4.588
18,19,20,22
T.G.T.G
1.982
17,18,20,21
T.A.T.A
-4.588
18,19,20,21
A.C.A,C
1.982
18,21
TT
-4.533
Table 7: Top 20 positive and negative Faith-Shap interactions in inDelphi from SHAP zero when predicting the frameshift frequency of DNA target sequences. GCGC and TCTC interactions at positions 17-18 and 21-22 and 18-19 and 21-22, respectively, show high-order microhomology patterns.
32

---

Top positive SHAP values
Top negative SHAP values
Position
Amino acid
Average SHAP value
Position
Amino acid
Average SHAP value
8
L
0.074
8
D
-0.046
4
L
0.074
4
E
-0.044
4
I
0,060
4
D
-0.044
6
C
0.058
5
P
-0.041
5
F
0.058
4
P
-0,040
8
V I M
0.053
4
K
-0,039
8
0.050
4
N
-0.039
8
0.050
4
-0,037
4
M
0.049
8
E
-0,036
F
2
I F
0.046
8
-0.036
5
0.042
9
P W
-0.036
0
0.040
7
-0.035
5
Y
0.037
9
W
-0,035
7
V
0.037
5
D P
-0.034
1
M
0.035
8
-0.034
3
K
0.034
4
Q
-0.033
4
V
0.034
6
W
-0,032
4
A
0.032
8
N
-0.031
9
K
0.032
8
K
-0,031
0.032
5
-0.030
Table Top 20 positive and negative SHAP values from SHAP zero when predicting protein fitness of protein sequences in Tranception. Top SHAP values are consistent with secondary structure predictors and agreement with KernelSHAP.
33

---

Top positive SHAP values
Top negative SHAP values
Position
Amino acid
Average SHAP value
Position
Amino acid
Average SHAP value
5
0.012
8
D
-0.108
4
L
0.008
4
D
-0.107
6
C
0,007
4
E
-0.103
8
L
0.006
8
E
-0.101
1
M
0.006
4
P
-0.100
2
F F
0.005
4
N
-0.100
0
0.005
8
P
-0.099
9
K
0.005
4
K
-0,096
3
K
0.003
5
P
-0,095
V
0.003
-0.095
8
G
-0.094
8
-0.093
8
N
-0.093
5
-0.092
8
W
-0.090
4
Q
-0.090
5
E
-0,089
6
W
-0.082
8
R
-0,080
4
G
-0.080
Table Top positive and negative SHAP values from KemelSHAP in Tranception. KerneSSHAP is biased toward negative SHAP values due to marginalization in training data.
34

---

Top positive interactions
Top negative interactions
Positions
Amino acids
Average interaction
Positions
Amino acids
Average
interaction
4,6
R.Y
0.013
4,6
K,E
-0.017
4,6
K,W
0.011
2,3
F.W
-0.011
5,9
K.P
0.011
4,6
G.F
-0.011
5,9
D.W
0.011
5,9
K,K
-0.011
2,3
F.K
0.010
7,9
V.P
-0.010
5,7
P.N
0.009
5,9
WII
-0.010
4,6
LE
0.009
4,6
K.Q
-0.008
4,6
K.Y
0.008
5,7
K.L
-0.008
5,9
F.K 0.008
4,6
Y.T
-0.008
2,3
F.R 0.008
2,3
F.F
-0.008
5,7
W.G
0.007
2,3
F.I
-0.008
4,6
K.F
0.007
5,9
E,Q
-0.008
2,3
FQ
0.007
5,7
K.I
-0.008
4,6
N.W
0.007
5,7
K.M
-0.008
4,6
G.H
0.007
5,7
A.N
-0.007
5,7
K,E
0.006
5,7
PF
-0.007
7,9
N.P
0.006
5,9
K.D
-0.007
5,7
K.Q 0.006
5,7
K.V -0.007
5,9
K.F 0.006
4,6
W.T -0.007
7,9
P.P
0.006
5,7
G,C
-0.007
Table 10: Top 20 positive and negative Faith-Shap interactions from SHAP zero when predicting protein fitness in Tranception reveals epistatic interactions in secondary structure predictors.
35

---

.
G
0.10
0.10
0.8
0.8
-010
.0.00
Targel INCUAATTE praetion
Targel INNQuance ponsion
Figure 5: Top interactions in TIGER. a, Top 80 Faith-Shap interactions in TIGER with SHAP zero and b, SHAP-IQ. Although overlapping interactions in SHAP zero and SHAP-IQ are in agreement, SHAP zero interactions concentrated around the seed region.
36

---

FS GINg (epraooo 120 Sto0 AND
Fath Ohop rntrrction
Tangal sequartce poatton
Tangel uequarice posticon
Figure 6: Top interactions in inDelphi. a, Top Faith-Shap interactions in inDelphi with SHAP zero and b. SHAP-IQ. While overlapping interactions are in agreement, the majority of SHAP-IQ interactions are not centered around the cut site.
37

---

0.08
0.06
.A.C.O
11.44
0.0
108 USRSAA
0.00
002
SHA value
Tou suttiru
0.00
-002
-0.04
-204
FardSHAP
-0.00
-006
DeepSHt
Tangel INqquMTTIN poaston
Target auqqeaoco Bax
suquarooss explereed
Figure 7: FastSHAP and DeepSHAP TIGER. FastSHAP and DeepSHAP estimates 1038 query sequences, with random subset of estimates shown for clarity. b, Total runtime of SHAP zero, FastSHAP, and DeepSHAP
1038 target sequences. While both methods are faster than SHAP zero, FastSHAP estimates do not correlate well with KemelSHAP, and DeepSHAP estimates not black-box and do not support amortized inference.
38

---

SHAP 100 FaNEHHF
16-
TOW sutket rACOTTN
1.0
10"
00
10"
-06
Tangal aequaeTcI poadtonn
Figure 8: FastSHAP on inDelphi. a. SHAP values of 40 random target sequences from FastSHAP. Total runtime of SHAP zero and FastSHAP over 134 target sequences. While FastSHAP faster than SHAP zero, Fast estimates do not correlate well with KernelSHAP.
39

---

66
04
-00
Figure 9: FastSHAP on Tranception. a, SHAP values on held-out sequences from FastSHAP. b, Total runtime of SHAP zero and FastSHAP over 200 sequences. While FastSHAP faster than SHAP zero, SHPP estimates do not correlate well with KerneSSHAP.
40

---

F1 Verifying Epistatic Interactions in Tranception
We evaluated whether the top five positive and negative interactions identified Table 10 correspond to epistasis in Tranception. As computing ground-truth Faith-Shap interactions is too computationally expensive at the scales we run SHAP zero, we opted use SHAP zero to predict the top mutations in sequences that would result in epistasis.
From the held-out sequence set (Appendix G.S first selected sequences containing the relevant amino acid pairs. For each pair, we use SHAP zero's 's Möbius transform to predict whether mutations the amino acids pairs will results in epistasis, where second-order Möbius coefficients quantify pairwise interactions. We extracted the top three mutation pairs per interaction and introduced these mutations into the selected sequences.
To measure epistasis, we compared the Tranception scores of the original selected sequence x", the corresponding single mutants x and x positions i and j. respectively, and the double mutant x Epistasis values were computed as [60]:
c= fx f(x)) f(x)) + f(x").
where f(+) denotes the Tranception score. By construction, c is equivalent to the ground-trute second-order Möbius coefficient for the corresponding amino acid pair.
Table 1I reports the average epistasis value found in Tranception for these interactions. Here, see that SHAP zero correctly identifies interactions where Tranception exhibits positive and negative epistasis.
Top positive interactions
Top negative interactions
Positions
Amino acids
Epistasis
Positions
Amino acids
Epistasis
4,6
R.Y
0.04
4,6
K,E
-0.04
4,6 K.W
0.03
2,3
F.W
-0.01
5,9
K,P
0.04
4,6
G.F
-0.02
5,9
D.W
0.04
5,9
K,K -0.05
2,3
F,K
0.02
7,9
V.P
-0.02
Table 11: Top positive and negative pairwise interactions identified by SHAP zero, with epistasis values from Tranception.
G Experimental Setup
Experiments that involve querying TIGER or Tranception were run on a single NVIDIA RTX A6000 machine. All other experiments were run on CPU.
G.1 Implementation of Computing the Sparse Fourier Transform
In all experiments recover Fourier coefficients, we applied the sparse Fourier algorithm q-SFT [40] through their GitHub repository https://gihhub.commaaaisssalaali
We specify which and n in each experimental subsection. There three tunable parameters in SHAP that determine the number of samples used: subsampling dimension b, number of subsampling matrices (C), and number of offset matrices (Pi). Given length-n sequence, these parameters determine the number of samples SHAP zero takes to recover the most significant Fourier coefficients: samples = g² x C P1 (n+ 1). Increasing b increases the number of samples SHAP zero takes exponentially, The theory of compressed sensing tells us that the quality of recovered top Fourier coefficients depends on the total number of samples queried, meaning that performance improves monotonically with respect to all parameters. We also observe this in our empirical results: increasing b results recovering more Fourier coefficients and better SHAP zero accuracy. Therefore, to determine the number of samples to take in SHAP zero, we maximize b as large as computationally feasible and P 3 (which follows the default parameters of g-SFT), ensuring
while leaving C the recovery of the top Fourier coefficients. All other parameters in SFS were left default,
41

---

which includes setting query_method method to "complex", sedlyssmmehare channel to "nso".
inehhod_soure to "identity", and
After determining the number of samples to use, we then optimized the hyperparameter noise_ sd, which sets the peeling threshold recover Fourier coefficients. To find the optimal value of noise_sd, sd. we performed coarse search from 0 to 20 at 20 evenly spaced points and used the recovered Fourier coefficients each level of noise_sd to predict the values of a test set of 10,000 samples (done by setting the test argument n_samples to 10, 000), After finding the general range where It² was the largest, we refined our search within that general range at a smaller step size. We report the specific hyperparameters used for q-SFT in each experimental subsection.
G.2 Comparison With Baselines
Computing exact SHAP values or interactions for q-ary functions requires (gg)) computations (see Appendix E.3 [.4). For realistic values of q and n in sequence models, even single evaluation is intractable; for example, estimating one SHAP value for inDelphi would take on the order of o years on our servers. As result, we benchmark SHAP zero against KerneISHAP and SHAP-IQ. two methods that yield unbiased Shapley estimations [1 25 62 65 To ensure fair comparisons, we conduct a thorough convergence analysis for both KemelSHAA and SHAP-IQ (see below), and verify that that both baselines produce stable explanations. In addition, we performed detailed analysis of SHAP zero's explanations and uncovered interactions that align with known biological motifs from the literature. Together. these results demonstrate that SHAP zero produces biologically plausible and scalable amortized explanations for sequence models.
G.3 Implementation of KernelSHAP, SHAP-IQ, and FastSHAP
To create the KernelSHAP baseline, we used the Python package SHAP's KemelSHAP implementa- tion. For the background set, we used training sequences from inDelphi and TIGER. In Tranception, we used deep mutational scanning GFP data 449 from ProteinGym [66). and filtered sequences that were mutated outside of the 10 sites chosen. In Tranception, we were able to run SHAP on the full background dataset, due to the manageable number of sequences present the dataset. However, in the instance of TIGER and inDelphi, we were unable to fcasibly run SHAP with all training sequences. Therefore, to determine the size of the background dataset, we ran KernelSHAP over different background dataset sizes on a subset of the held-out sequences and plotted the R of the KernelSHAP estimates with respect to the largest background dataset size. Error bars are generated by computing the R² over each sequence's 's KernelSHAP estimates. We picked the background dataset size where KernelSHAP values converged with an R² > 0.97 to run on all the held-out sequences.
To compute Shith-Shap interactions, we used SHAP-IQ's implementation, available on their GitHuub repository
Since SHAP zero's recovered interactions were greater than 3 in both TIGER and inDelphi. we wanted to run SHAP-IQ at max_ __rderrdrrrrr (f) > 3. However, we ran into computational issues in both cases, resulting in us setting = 3. In Tranception, we set L = 2, since SHAP zero wasn't 't unable to recover significant interactions greater than Additionally, since SHAP-IQ was computationally too expensive to run over all sequences, we randomly selected few sequences to analyze. We used the same background dataset from KemneSSHAP and hyperparameter tuned budget (which determines the number of model evaluations to make). Similar to KerneISHAP, we ran SHAP-IQ over different budgets on subset of held- out sequences and plotted the R2 of the SHAP-IQ interactions with respect to the largest budget size. Error bars are generated by computing the R² over each sequence's Faith-Shap estimates, stratified by interaction order. We picked the budget where Faith-Shap estimates converged with an R² > 0.6 to run all the held-out sequences. In the case of Tranception, we noticed that the Faith-Shap interactions converged quickly to I²2 to use budget 10, 000 for our experiments. We report the specific KemelSHAP and Faith- Shap parameters for TIGER and DnDelphi in each experimental subsection. Fig. 10 shows the convergence plots for all three models, with dashed lines showing what experimental parameters were used.
at budget = 10, 000. Therefore, we opted
To run FaP we used the FastSHAP implementation available on their GitHub repository (https://github.cc http://aaooo/e///stttiioooaoo
We removed all instances of calling the softmaa function. We replaced the KI. divergence loss used to train the surrogate neural network for mean squared error loss since the FastSHAP implementation on their GitHub is designed for classification and not
42

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40 (TSEE))
1.0
Fath-Smppp (TIGER)
1.00
0.
0.90
0.
0.00
10
10000
20000
30000
40000
60003
60000
Baddggoound
Buudge
Horel9HA (n0epm0))
1.0
h- R7 (nDeaht)
0.75
0.50
0.00
Faedeyooun comple
Faudge
Tarnetee (treroppinn)
Falh Blup (Tunnatoon)
0.98
0.04
0.9967
30000
50002 E0000
Fladegiound cample sine
Busgge
Figure 10: Shapley convergence plots in TIGER, inDelphi, and Tranception. We hyperparameter tuned the size of the background dataset KernelSHAP and budget in SHAP-IQ. Dashed lines indicate what experimental parameters were used.
regression. Following the FastSHAP implementation left on their GitHub for the census dataset, the surrogate model is implemented using a neural network consisting of three fully connected layers, where each layer is 128 units wide, with ReLU activation. We did not train the surrogate model according to their CIFAR implementation because their CIFAR-IO surrogate model uses pre-traine mre-trained ResNet-50 model designed for image-specific tasks. The surrogate model trained the entire training set. All other FastSHAP parameters were left as default according to their GitHub implementation on the CIFAR-10 dataset.
G.4 Calculation of Top SHAP Values, Top Faith-Shap Interactions, and Total Runtime
To determine the top SHAP values the Appendix (see Tables 2 3
6 and we separated the
SHAP values into positive and negative values. For the positive and negative grouping. we calculated the average contribution of each nucleotide at each position to report the average SHAP value. The same procedure was used to determine the top Faith-Shap interactions (see Tables but instead by calculating the average contribution computed by summing the sample and computational complexities of each algorithm. We estimated the runtime per sample by dividing the total runtime by the number of query explanations performed.
7 and TOP. each unique interaction. Total runtime was
G.5 Visualization of SHAP Values and Faith-Shap Interactions in TIGER and inDelphi
To visualize SHAP values (see Fig. 2. Fig. 3P. Fig. 7p. and Fig. Kh), randomly sampled 50 sequences and plotted their SHAP values on scatterplot. To plot histogram of Faith-Shap interactions (see Fig. 2 and Bo), we used the visualization scheme described in (28) After separating the positive and negative interactions, we evenly distributed the interaction magnitudes across all affected nucleotides. We reported the summed contributions of each nucleotide each position on bar graph. However, slightly deviating from (28), rather than stratifying the bar plots by the order interactions, we stratified them by nucleotide type and only plot interactions with an order greater than 1. For more details, refer to Appendix B of (28). To visualize Faith-Shap interactions as shown the Appendix (see Fig. 5 and we plotted interactions as data points with line connecting points to show the order of interaction. For visual clarity, we only plotted the larger interaction if interactions with the same nucleotides the same positions have an estimated difference of 0.001. We plotted the top 80 interactions from SHAP zero and SHAP-IQ.
43

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G.6 TIGER
We considered the sequence-only implementation of TIGER by inputting length-n 26 perfect match target sequences, where the model assumes that the gRNA is perfect sequence complement of the target sequence. We set q = 4, n = 26, and range where R² was the largest, we refined our search using step size of 0.025. We used h = 7 and noise_sd = 0.725, which corresponded to an R of 0.55.
7. For noise_ sd. after finding the general
We used the held-out perfect match target sequences provided by TIGER 33 to predict the experi- mental scores. We evaluated the performance of the recovered Fourier coefficients using Pearson correlation, Spearman correlation, area under the receiver operating characteristic curve (AUROC), and area under the precision-recall curve (AUPRC). AUROC and AUPRC were used because TIGER allows thresholding predictions to formulate binary classification problem. We created our linear and pairwise interaction model baselines using sklearn in Python. Our baseline models were trained on one-hot encoded sequences using the training data from TIGER to predict the experimental guide score, employing Ridge regression with 0.5. To construct the pairwise interaction model, we used sklearn's 's PolynomialFeature, implementation, setting degree to 2 and interaction_onlyy to True.
For KernelSHAP and SHAP-IQ, background samples were randomly drawn from the training set. During hyperparameter tuning, we evaluated KernelSHAP using 10, 15, 30, 40, and 50 background samples across 30 held-out sequences; we selected 30 background samples for our experiments. For SHAP-IQ, we performed a hyperparameter search over budgets of 1,000, 10,000, 30,000, 50,000, 60,000, and 70,000 on two held-out sequences; we chose a budget of 50,000 for downstream analysis. Due to SHAP-IQ's 's high computational cost, we limited evaluation to a randomly selected subset of eight sequences. We additionally set max __nderrrrr__rer = 3: the highest order feasible before encountering computational constraints. SHAP zero was run on all held-out sequences.
DeePSHAP was implemented using the Python package SHAP. The background dataset consists of 5000 samples randomly drawn from the training set, which is the number of background samples TIGER utilizes in their own DeepSHAP implementation. All other parameters were left as default.
G.7 inDelphi
We applied SHAP zero inDelphi by inputting 40 DNA target sequences over a 4 alphabet, which were additionally padded with 20 nucleotides on each end due to inDelphi's minimum length requirement, get their frameshift frequency. We refined our noise_ sdd estimate using step size of 0.1. We used b- = and noise sd = 15, which corresponded to an R of 0.82.
We used the held-out DNA target sequences provided by inDelphi (34 to predict the experimental scores. We evaluated the performance of the recovered Fourier coefficients using Pearson and Spearman correlation. Since inDelphi does not have option to threshold predictions like in TIGER, we do not report AUROC or AUPRC. The bascline models were trained on one-hot encoded sequences using the training data from inDelphi to predict inDelphi's 's predicted frameshift frequency, as the experimental frameshift frequencies were not provided. KernelSHAP, SHAP-IQ, and FastSHAP were created using the same procedure in the TIGER experiment.
SHAP zero was run on all held-out sequences plus an additional 50 training sequences due to the limited amounts of held-out data to compute SHAP values and Faith-Shap interactions. For hyperparameter tuning, we evaluated KerneÍSHAP using 5, 10, 15, 30, 40, and 50 background samples across 10 held-out sequences, and selected 30 background samples for our experiments. To hyperparameter tune SHAP-IQ. we performed a hyperparameter search over budgets of I, 1,000, 5,000, 10.000, 20,000, 25,000, 30,000 one held-out sequence, and chose budget of 20,000 for experiments. We limited our final SHAP-IQ experiments to run on a randomly selected subset of four sequences. Similar TIGER, we additionally set max max_order order = 3 due to computational constraints.
G.8 Tranception
Unlike TIGER and inDelphi, Tranception takes in protein sequences. We consider the "large" model on Tranception with retrieval, which has 700 million parameters 35 Tranception with retrieval scores each mutant by taking a weighted average of two values: the log- likeliill from the main transformer model and the likelicel derived from the multiple sequence aligment (MSA) at the
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mutant's position, computed from the empirical amino acid distribution. When applying SHAP zero on Tranception, the MSA is initialized once during the beginning and remains fixed throughout the duration of sampling, ensuring consistent sequence-fune funncionn model.
We feed in the cannonical wildtype GFP sequence (which be found at [66] and mutate the 10 epistatic sites [49] experimentally tested (located in Supplementary Table 1). We additionally
use the MSA provided from ProteinGym [66] for retrieval. We refine our noise_ sd estimate using step size of 0.01. We used b and noise_sd sd = 0.03, which corresponded to an It2 of 0.97.
SHAP zero evaluated 200 held-out sequences, randomly generated by introducing up five mutations from the wildtype. To ensure comprehensive coverage, each amino acid appeared at least at every position across the held-out set. We opted for synthetic test sequences because the deep mutational scanning (DMS) dataset from [49 lacked full coverage of all amino acids across all positions, which would have limited the effectiveness of SHAP zero to find epistatic interactions. KernelSHAP and SHAP-IQ are run using the DMS dataset from 44), containing 79 sequences, over the sites chosen. Since n 10, we were able run KerneISHAP and SHAP-IQ over all of the DMS data without computational issues. We run SHAP-IQ with max_order 2 and budget = 10, 000 over 50 held-out sequences.
To create the linear and pairwise baselines, our baseline models were trained on our 200 sequence dataset predict Tranception's 's score. We opted to use these sequences specifically, Tranception was trained in an sulf-supe fashion, and does not have any explicit sequence-functinn training data unlike TIGER and inDelphi. We then tested SHAP zero, linear, and pairwise models predict experimental protein function from the DMS sequences.
G.9 Visualization of SHAP Values and Faith-Shap Interactions in Tranception
To visualize SHAP values (see Fig, 4 and Fig. 92), we computed the average SHAP value per amino acid at each position, and plotted the top 20% of the positive and negative SHAP values on a heatmap. In the case of KernelSHAP. we plotted all the positive SHAP values the heatmap, since we observed that KernelSHAP was biased toward negative values (see Discussion). To visualize Faith-Shap interactions (see Fig. 4e), we computed the average pairwise interaction of amino acids. We plotted the top 20% of the positive and negative interactions on the heatmap.
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Reviews:
[{"review_id": "jHIbrgsMrx", "reviewer": "Reviewer_jhji", "summary": "This work develops a new approach for SHAP values estimation in black box biological sequence models, focusing on amortisation of computational cost. The approach is to, instead of querying SHAP values for every sequence of interest, build a global sparse representation of the model, which only needs to be done once, using a sparse Fourier transform, and connect this to sequence-specific SHAP values via a Möbius transformation. They test the method's ability to explain guide rNA binding (with TIGER), DNA repair outcome (with inDelphi), and epistasis in protein language models (with Tranception). They show good agreement with SOTA approaches, both at extracting SHAP values and high-order interactions, at a much lower computational cost, especially in the regime of small sequences and large number of queries. The article is well written and clear.", "questions": "1. If I understand correctly, you are querying variation on 10 positions known to be epistatic in GFP. The first question is how does your Fourier representation correlate with experiments, compared to Tranception, linear and pairwise representations (an equivalent to 2b and 3b)? Maybe I missed it. \nI think that perhaps a more interesting question than looking at known interacting positions would be to, given many positions, identify the ones that are epistatically interacting. This would be more in line with tasks such as identifying allosteric sites involved in ligand binding, protein design, etc. Would this be computationally viable? (you would increase sequence length but would look at a sparser space) See question 2, related.\n2. How far in sequence length can you take this approach? I think a scalability analysis in a protein and RNA/DNA setting would be useful. \n3. Have you considered using other sparse representations such as wavelets instead of a sparse Fourier basis? What do you think would be the relative pros and cons? \n4. If I understand correctly, in line 223 you say that the overlapping high-order feature interactions between SHAP zero and SHAP-IQ correspond to 95% of the total SHAP zero high-order feature interactions and 5% of those from SHAP-IQ. Why the difference in total number of high-order interactions between these two approaches? Also, why is it the opposite in the DNA repair example, line 244?", "limitations": "It would be useful to have a better picture of the scalability of the approach as mentioned in question 2.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\nInteresting idea, well exposed, with potential to be portable across multiple tasks with biological sequence models and have therefore broad adoption. The approach is technically solid and the experiments are for the most part sufficiently appropriate to be convincing of this strategy's value. I find it particularly interesting that the Fourier coefficients recovered from SHAP zero perform so well at the RNA binding and DNA repair tasks, strongly suggesting that this is a meaningful representation of the data. \n\nWeaknesses:\nThe approach seems to have benefits in computational costs whenever the number of queries is large, the function sufficiently sparse, and the length of the input sequence is not too large. This questions how broadly adopted this strategy can be in practice. \n\nI found that the protein language model experiment could have been taken further to be more representative of the kinds of tasks the community is interested in (see questions).", "was_revised": true, "final_justification": "All issues were resolved. I recommend the authors to add a small discussion on scalability of the approach to the article, in line with the rebuttal. I am keeping the original acceptance score.", "rebuttal": "We thank reviewer jhji for their constructive feedback. Below, we address specific questions: \n\n> how does your Fourier representation correlate with experiments compared to Tranception, linear and pairwise representations\n\nTo address this question, we performed new experiments comparing SHAP zero against Tranception and linear/pairwise baselines on the deep mutational scanning data from [1], following the procedure in Appendix E.6:\nModel|Pearson|Spearman\n-|-|-\n |Tranception|0.57|0.75\n |SHAP zero|**0.55**|0.69\n |Linear|0.48|0.54\n |Pairwise|0.33|0.41\nSHAP zero outperforms the linear and pairwise baselines in generalizing to deep mutational scanning data. It also closely approximates the performance of the full Tranception model. We will add these results to our protein experiments in Section 4. \n\n> I think that perhaps a more interesting question than looking at known interacting positions would be to, given many positions, identify the ones that are epistatically interacting… Would this be computationally viable? (you would increase sequence length but would look at a sparser space)...  How far in sequence length can you take this approach?\n\nWe agree that identifying epistatic positions at scale is an exciting future direction that SHAP zero is well-equipped to handle. The scalability of this approach depends both on the query speed and the relative sparsity of the model. On our single GPU setup, assuming a time budget of 10 days and that the relative sparsity of the model remains constant as we scale up $n$:\n- We can recover interactions across approximately $n=30$ sites for larger, slower-to-query models like Tranception.\n- For smaller, faster-to-query models like TIGER, the same budget would allow us to scale to $n=100$ sites.\n\nAs $n$ grows, we speculate that the relative sparsity level will likely decrease, enabling SHAP zero to analyze significantly more positions than we report here.\n\nWe additionally clarify that, in our current experimental setup, finding epistatic interactions given a subset of positions is an important and high impact problem [2-4], especially in the context of discovering proteins with novel biological motifs. Screening just 10 positions results in a combinatorial space of over 10 trillion sequences ($20^10 \\sim 1 \\times 10^13). Existing methods screen this landscape by restricting their search to small, local neighborhoods within the protein's 3D structure [2, 4] or over a small fraction of amino acids [3]. We emphasize that SHAP zero, unlike these methods, does not require such restrictive assumptions, which enables the recovery of epistatic interactions previously inaccessible at scale.\n\n> Have you considered using other sparse representations such as wavelets instead of a sparse Fourier basis?\n\nWe have not considered other representations such as wavelets. In the context of biological sequence-function models, Fourier is the only standard basis which defines interactions in sequences, consistent with the definition of epistasis in statistical genetics [5]. SHAP zero could, in principle, be generalized to different bases, which would be an interesting future direction.\n\n> Why the difference in total number of high-order interactions between [SHAP zero and SHAP-IQ in the protein experiments]? \n\nBoth SHAP zero and SHAP-IQ are approximating Faith-Shap interactions in Tranception. Given the computational budgets provided to both algorithms, we observe that SHAP zero identifies more biologically meaningful interactions. We attribute this success to SHAP zero’s underlying assumption of Fourier sparsity, a principle that previous studies suggest holds for biological fitness landscapes [6-8]. This assumption allows SHAP zero to structurally subsample the model to efficiently isolate key interactions. In contrast, SHAP-IQ's reliance on random feature subset sampling is less efficient for this problem and requires a larger computational budget to achieve comparable results.\n\n> Also, why is it the opposite in the DNA repair example, line 244?\n\nThank you for pointing out the inconsistency in the DNA repair example—that is a typo on our part, and we apologize for the error. The trend is the same as in our protein experiments. The corrected text should read:\n“interactions (constituting 85% and 3% of the total interactions in **SHAP zero** and **SHAP-IQ**, respectively)”\n\n[1] Sarkisyan et al. \"Local fitness landscape of the green fluorescent protein\" (2016)\n\n[2] Weinstein et al. “Designed active-site library reveals thousands of functional GFP variants” (2023)\n\n[3] Wu et al. “Adaptation in protein fitness landscapes is facilitated by indirect paths” (2016)\n\n[4] Starr et al. “Epistasis in protein evolution” (2016)\n\n[5] Poelwijk et al. \"The context-dependence of mutations: a linkage of formalisms\" (2016)\n\n[6] Aghazadeh et al. “CRISPRLand: Interpretable large-scale inference of DNA repair landscape based on a spectral approach” (2020)\n\n[7] Brookes et al. \"On the sparsity of fitness functions and implications for learning\" (2022)\n\n[8] Tsui et al. “On recovering higher-order interactions from protein language models” (2024)"}, {"review_id": "zLQ4GhbaRc", "reviewer": "Reviewer_hhqE", "summary": "The authors describe a method for computing Shapley values for biological sequence-to-function models with query-only access that scales significantly better than existing exponential-time methods. It achieves this by paying an upfront polynomial-time cost to create a global Fourier sketch of the model, which is tractable for biological sequence contexts given their sparsity. The authors then show that for a given sequence, the Fourier coefficients can be mapped to Shapley values via a Mobius transform that scales with O(q^l), where q is the dimensionality of the input space (e.g. 4 for nucleic acids and 20 for proteins) and l is the maximum order of interactions considered. To do this, the authors introduce a q-ary extension of their previously-published binary Mobius transformation. The authors test their method on three existing sequence-to-function models (TIGER, inDelphi, and Transception). For TIGER and inDelphi, they show that their Fourier coefficients predict the test set better than linear or pairwise models and that their calculated Shapley values correlate well with those produced by KernelSHAP and SHAP-IQ while being orders of magnitude faster. For Transception, the authors additionally found epistatic interactions between certain residues of GFP.", "questions": "How does SHAP zero perform when asked to evaluate rare and highly deleterious mutations? Even if it doesn’t assign as highly-negative of values as a method like KernelSHAP, does it still correctly identify those mutations as being the most explanatory?\n\nHow does SHAP zero perform in the context of a sequence landscape that would be computationally-intractable for methods like KernelSHAP and SHAP-IQ? Suggestions include models trained on MPRA datasets or even a synthetic dataset where all the feature contributions and higher-order interactions are known. Even though it would be impossible to compare to slower methods, it would add to the paper’s significance if the method was able to pull out known feature contributions/interactions while also identifying new ones.\n\nThe Tranception case study is the least fleshed-out and most confusing, especially for epistatic interactions. The statement in the 4d figure legend “SHAP zero reveals numerous epistatic interactions with Proline (P) and Lysine (K).” is not actually shown in the figure (that I can tell) and needs to be inferred from Appendix D tables. It’s also unclear why the correlations with SHAP-IQ values are relatively low, and there is no analysis of whether the Fourier coefficients predict mutational effects better than a linear or pairwise baselines, so while SHAP zero is pulling out interactions, it’s hard to evaluate how trustworthy those conclusions are.\n\nModel evaluation cost:  The initial sketch requires many queries for TIGER/inDelphi (appendix). For large transformer models whose forward pass dominates runtime, the wall‑clock break‑even point (e.g. 25 k sequences) may be restrictive for some users. Discussion of adaptive or incremental sketching could be beneficial.", "limitations": "Yes, the authors adequately addressed the limitations of their work, except for some points that were addressed earlier.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- The paper was clear and well-motivated.\n- The authors benchmarked their method in three different biological contexts, covering RNA, DNA, and protein sequences.\n- Their q-ary Mobius transform allows for amortization in biological sequence contexts, filling in a “missing link”, so to speak.\n- The paper compares to FastSHAP, which uses a neural network to amortise, and provides some insight into why FastSHAP does not work very well for biological sequence models.\n- SHAP zero is impressively fast while maintaining decent-to-great correlation with slower methods, opening the door to interpreting sequences-to-function models that might have been intractable for slower methods.\n- Theoretical analysis of the approach is provided, filling in details such as complexity.\nWeaknesses:\n- The sparse Fourier algorithm assumes equal probability of all feature subsets, which doesn’t reflect real sequence distributions. The authors recognize this as a limitation when they note that KernelSHAP has more negative values for rare token values, compared to SHAP zero, but they don’t fully flesh out the implications for applying their method since rare nucleotides/amino acids usually reflect strong negative selection. For example, it would be helpful for potential users to know how their method labels stop codons/CpGs/etc to aid in interpreting results.\n- The major advantage of SHAP zero is the ability to assign Shapley values in contexts that were previously intractable, so it seems like a missed opportunity that the authors don’t apply their method to any such context since that’s where future users of the tool will likely want to use their method (the Transception case study is still in the context of a small GFP landscape).", "was_revised": true, "final_justification": "I maintain my score having read the author's comments. Technically solid, high impact in one sub-area of AI, but not wide ranging enough to get a 6.", "rebuttal": "We thank reviewer hhqE for their comments. Below, we address the following questions:\n\n> How does SHAP zero perform when asked to evaluate rare and highly deleterious mutations? Even if it doesn’t assign as highly-negative of values as a method like KernelSHAP, does it still correctly identify those mutations as being the most explanatory?\n\nWe clarify that Shapley estimation algorithms finds the most impactful mutations on a sequence, which includes both deleterious and beneficial mutations. While KernelSHAP tends to assign more negative values, the overall explanations SHAP zero returns is sufficient to identify rare and impactful mutations. SHAP zero’s explanations achieve a correlation of $\\rho=0.97$ with KernelSHAP’s estimates, indicating that both methods agree on the relative importance of mutations. Examining the top mutations (Tables 8 and 9), SHAP zero identifies all the top positive mutations and 17 out of 20 of the top negative mutations found by KernelSHAP. The three negative interactions that were missed were all ranked in the bottom five of KernelSHAP's list. This demonstrates that while SHAP zero's negative value assignments are more conservative, it is still able to capture the most impactful mutations.\n\nWe will add a summary of this to the Discussion section.\n\n> How does SHAP zero perform in the context of a sequence landscape that would be computationally-intractable for methods like KernelSHAP and SHAP-IQ? Suggestions include models trained on MPRA datasets or even a synthetic dataset where all the feature contributions and higher-order interactions are known. Even though it would be impossible to compare to slower methods, it would add to the paper’s significance if the method was able to pull out known feature contributions/interactions while also identifying new ones.\n\nSHAP zero will be able to recover meaningful interactions in landscapes that are intractable for methods like KernelSHAP and SHAP-IQ under the key assumption of Fourier sparsity. This means that only a small fraction of interactions are relevant to the model’s output. In the context of models trained on MPRA datasets, previous studies show that such models encode regulatory motifs which are known to be sparse in Fourier [1-2], suggesting that our method's core assumption holds in this domain. \n\n> it’s unclear why the correlations with SHAP-IQ values are relatively low… how trustworthy those conclusions [(SHAP zero on Tranception)] are… [with] linear or pairwise baselines.\n\nWe clarify that the correlation between SHAP zero and SHAP-IQ is $\\rho=0.61$, which demonstrates that the interactions recovered are in agreement with one another to some degree. Our analysis of the top biological motifs from Tables 8 and 10 shows that the top interactions recovered by SHAP zero corresponds to known motifs that affect protein stability. Regardless, to better assess the trustworthiness of SHAP zero, we have additionally performed two new experiments. First, we took the top 5 positive and negative interactions identified by SHAP zero from Table 10 and explicitly verified that these interactions are epistatic in Tranception. Second, we compared the performance of SHAP zero’s recovering Fourier coefficients against Tranception and linear and pairwise baselines (see Reviewer jhji for the full Table). We observe that SHAP zero ($\\rho=0.55$) is able to closely approximate the performance of Tranception ($\\rho=0.57$), suggesting that the epistatic interactions SHAP zero captures are biologically meaningful. \n\nWe will add these new results to Section 4 and the Appendix and update Fig. 4 to better highlight the epistatic interactions discovered by SHAP zero. \n\n> Model evaluation cost… may be restrictive for some users\n\nThe one-time cost required for SHAP zero's initial sketch is relatively low compared to the cost of training modern AI models. SHAP zero is designed to use less computational power in the long run by reducing the amortized cost of future explanations to near-zero.\n\nThe model evaluation cost of SHAP zero is primarily determined by two factors: the query speed and the Fourier sparsity of the model. In the case of ML-guided protein design, it is common to screen for a pre-selected list of important sites [3-4], rather than over the entire sequence. When running SHAP zero for larger, slower-to-query models like Tranception, this helps keep the problem well-defined and computationally feasible. For analyses over a very large number of sites, an interesting extension would be to run multiple, smaller SHAP zero instances across different regions of the sequence, and combine the explanations. This approach would be more computationally efficient than a single, large run because each SHAP zero instance would only need to recover a smaller set of sparse Fourier coefficients.\n\nWe will additionally release a pip package with the camera-ready version. This package will automatically tune SHAP zero’s parameters based on a user-defined sample budget (following our computational budget selection in Appendix E.1), providing the best possible Shapley estimates for their available resources.\n\n[1] Agarwal et al. “Massively parallel characterization of transcriptional regulatory elements” (2025)\n\n[2] Dalla-Torre et al. “Nucleotide Transformer: building and evaluating robust foundation models for human genomics” (2024)\n\n[3] Bian et al. “Optimizing enzyme thermostability by combining multiple mutations using protein language model” (2024)\n\n[4] Lipsh-Sokolik et al. “Addressing epistasis in the design of protein function” (2024)"}, {"review_id": "mgalhfBnv5", "reviewer": "Reviewer_b3je", "summary": "This paper introduces SHAP zero, an algorithm for explaining black-box biological sequence models. By amortizing the computational cost of calculating Shapley values across many queries, the method enables large-scale interpretability studies that were previously intractable. The main idea is to create a one-time global sketch of the model using a sparse Fourier transform, from which local explanations for future queries can be derived at near-zero marginal cost. The authors demonstrate its effectiveness on models for gRNA efficacy, DNA repair, and protein fitness.", "questions": "- In the analysis of the Tranception model, the paper notes the use of \"Tranception with retrieval\". This model's function is not a simple mapping from a sequence X to a real value R, i.e., f:X→R. Instead, it incorporates information from a MAS, making the function closer to f:(X,M)→R, where M represents the MSA. Could the authors clarify how this additional input from the MSA space was handled within the SHAP zero framework?\n\n- The validation is focused on DNA and protein sequence models. Could the authors comment on the potential applicability of SHAP zero to other types of biological data that also have a combinatorial nature but a different structure, such as protein-protein interaction networks or models of RNA-protein binding?", "limitations": "- The authors have provided a thoughtful discussion of the limitations of their work, particularly regarding the assumptions of Fourier sparsity and the uniform marginalization strategy. I find this discussion to be transparent and appropriate.", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n- The manuscript is well-written and presents an exciting application of model interpretability to the important domain of biological sequences. The connection established between Shapley values, the Möbius transform, and the sparse Fourier transform is elegant and provides a strong theoretical foundation for the proposed method.\n- The case study involving the Tranception model is interesting. The method demonstrates a promising capability to uncover the mechanistic basis of epistasis in Green Fluorescent Protein (GFP), a task of significant biological interest.\n\nWeaknesses\n- While the results are promising, the claims about capturing epistatic interactions would be substantially strengthened by including a broader range of examples. The paper presents a single case study on GFP using the Tranception model. To more robustly validate the method's ability to uncover epistasis, it would be beneficial to apply it to other proteins with well-characterized or computationally predicted epistatic networks\n- A fundamental challenge is the evaluation of the generated explanations. The manuscript demonstrates that the explanations align with some known biological phenomena, but the concept of a \"ground truth\" for interpretability remains elusive. How can we quantitatively assess the correctness of an explanation, especially in cases where the underlying biological mechanisms are not fully understood by human experts? Without a clear evaluation framework, the interpretations risk being perceived as plausible yet speculative hypotheses.", "was_revised": true, "final_justification": "My concerns are addressed, and I will be glad to see the XAI in the area of AI4Sci.", "rebuttal": "We thank reviewer b3je for taking the time to review our paper. We address the following questions:\n\n> Could the authors clarify how this additional input from the MSA space was handled within the SHAP zero framework (for Tranception with retrieval)?\n\nIn our framework, the multiple sequence alignment (MSA) is treated as a fixed component of the overall black-box model, not as a separate input for each query.\n\nTranception with retrieval scores each mutant by taking a weighted average of two values: the log-likelihood from the main transformer model and the log-likelihood derived from the MSA at the specific mutant's position (see Fig. 3 and Equation (4) in [1]). The MSA-derived likelihood is based on the empirical amino acid distribution at that position.\n\nWhen using SHAP zero, Tranception with retrieval is treated as a single, consistent black-box function. The MSA is initialized only once at the beginning and remains static for all subsequent mutant evaluations. The user does not need to re-initialize or change the MSA for every new mutant, ensuring our definition of the sequence-function model from the Introduction remains consistent.\nWe will add a summary of this discussion in the Appendix.\n\n> The validation is focused on DNA and protein sequence models. Could the authors comment on the potential applicability of SHAP zero to other types of biological data that also have a combinatorial nature but a different structure, such as protein-protein interaction networks or models of RNA-protein binding?\n\nThank you for your question. SHAP zero can be applied to other combinatorial biological problems. For problems such as protein-protein binding, the application is seamless: we would frame the input as the combined sequence of interacting proteins and the output as the binding affinity. Assuming this mapping is sparse in Fourier (i.e., a small fraction of residue-residue interactions contribute to binding), SHAP zero will generalize. For problems such as RNA-protein binding, an interesting extension of SHAP zero would be to compute the Fourier transform over different alphabet sizes ($q$=4 and $q$=20) using generalized $q$-ary Fourier algorithms [2]. \n\n[1] Notin et al. “Tranception: protein fitness prediction with autoregressive transformers and inference-time retrieval” (2022)\n\n[2] Tsui et al. “Efficient Algorithm for Sparse Fourier Transform of Generalized $q$-ary Functions” (2025)"}, {"review_id": "9hlGRDOeZE", "reviewer": "Reviewer_GnFL", "summary": "In this paper, the authors introduce SHAP zero, a fast approximate method for computing SHAP values for model explainability based on sparse Fourier transforms and near-zero marginal cost for queries over large biological sequence datasets. The method is evaluated on three sequence-based models for predictive tasks related to guide RNA efficacy, DNA repair, and protein fitness. SHAP zero shows comparable results to KernelSHAP, while being many times faster.", "questions": "The manuscript repeatedly refers to a model “sketch”, and specifies that the Fourier transform of a black-box model f is the “global sketch” of that model. What is the exact definition and meaning of “sketch” in this context?\nWhat is the rationale for excluding gradient-based attribution methods, considering that all of the models f in the work are differentiable neural networks?", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 1, "significance": 1, "originality": 2, "strengths_and_weaknesses": "Strengths\nThe method is mainly compared against KernelSHAP, and seems to give a significant speedup with comparable results for sequence-based models.\nModel explainability is an important and understudied problem for biological sequence models.\n\nWeaknesses\nThe notation for Shapley values is appreciated, but probably better suited to a Methods or Preliminaries section than almost immediately in the Introduction.\nThe main text would benefit from pseudocode showing the SHAP zero algorithm.\nOverall, the connections between the sparse Fourier transform, Mobius transform, and SHAP values are mathematically clear, but not intuitively clear. In addition to writing out the algorithm, it may be beneficial to simply explain how traditional SHAP and SHAP zero differ.\nThere is no mention of gradient-based attribution methods, like integrated gradients, which are possibly the most widely adopted and easy to use methods for explaining sequence-based models.", "was_revised": true, "final_justification": "The authors' response has addressed my comments and questions and I will increase my score accordingly.", "rebuttal": "We thank reviewer GnFL for their comments. We address the following questions:\n\n> The notation for Shapley values is appreciated, but probably better suited to a Methods or Preliminaries section than almost immediately in the Introduction. The main text would benefit from pseudocode showing the SHAP zero algorithm.\n\nWe will provide a more thorough explanation of Shapley values and add a SHAP zero algorithm box to the main text.\n\n> Overall, the connections between the sparse Fourier transform, Mobius transform, and SHAP values are mathematically clear, but not intuitively clear.\n\nBriefly, the connection between SHAP values, the Möbius transform, and the sparse Fourier transform lies in how we represent and recover feature interactions in complex models. SHAP values attribute a model’s prediction to individual features and their combinations by decomposing the function into additive contributions over subsets. This decomposition aligns with the Möbius transform, a mathematical framework that captures how a function defined over feature subsets can be expressed in terms of unique contributions from each subset. However, directly computing this is intractable for high-dimensional models. The sparse Fourier transform offers a computational shortcut by exploiting the empirical sparsity of interactions in biological systems—only a few feature combinations significantly affect the outcome.\n\nWe have made this connection clear throughout the paper, specifically around lines 52, 153, and 182. To address the reviewer's concern, we will add additional explanations to make the connection more clear.\n\n> The manuscript repeatedly refers to a model “sketch”, and specifies that the Fourier transform of a black-box model $f$ is the “global sketch” of that model. What is the exact definition and meaning of “sketch” in this context?\n\nWe formally defined the sketch as the $s$-sparse approximation of the Fourier transform of the model, as detailed in Equation (3). We refer to this as a \"global\" sketch because this Fourier representation describes the entire function's behavior, independent of any single input query.\n\nWe clarify that the term \"sketch\" is standard terminology in theoretical computer science and machine learning. The core idea of a sketch is to create a compressed summary of a model that captures the most relevant information. We are happy to make this more clear in the text. \n\n> What is the rationale for excluding gradient-based attribution methods, considering that all of the models $f$ in the work are differentiable neural networks?\n\nWe clarify that the focus of our work is on amortizing Shapley-based explanations in **black-box sequence models**. Gradient-based attribution methods, which rely on model internals, fall under white-box explainability methods. Regardless, we have addressed white-box explainability methods in the Discussion section, and additionally provided experiments in the Appendix on running DeepSHAP, another white-box method, on TIGER in Appendix D.\n\nWe emphasize that with the growth of large-scale AI sequence models with proprietary access, the choice of black-box explainability methods over white-box ones is more of a need than a luxury for biological discoveries."}]

Author rebuttal:

Meta-review: This paper introduces SHAP zero, an algorithm for explaining black-box biological sequence models. By amortizing the computational cost of calculating Shapley values across many queries, the method enables large-scale interpretability studies that were previously intractable. The main idea is to create a one-time global sketch of the model using a sparse Fourier transform, from which local explanations for future queries can be derived at near-zero marginal cost. The authors demonstrate its effectiveness on models for gRNA efficacy, DNA repair, and protein fitness.

The reviewers agreed that the paper is very well written, and that the methods presented are exciting and important, and also the core idea is neat and clever. The discussion clarified reviewers questions, and there the paper seems to be at the level suitable for this conference.
Decision: Accept (poster)