Studying Large Language Model Generalization with Influence Functions

Limitations

• The analysis is local and limited to pretrained models, MLP parameters, models up to 52 billion parameters, and a fraction of the pretraining corpus.Influence functions may not capture nonlinear phenomena such as complex circuit formation or global representational rearrangements, and the study evaluates their fit to the proximal Bregman response function rather than the ultimate training counterfactual.Source: §1 · p. 8

Method and validation

• EK-FAC makes influence estimates competitive with LiSSA while being significantly faster, enabling analysis of transformer language models up to 52 billion parameters.The approach also uses query batching to share training-gradient computation across dozens of influence queries.Source: §1 · pp. 5, 8

Influence structure

• Influence is heavy-tailed yet distributed across many sequences, suggesting typical behaviors do not arise from direct memorization of only a handful of examples.The influence distribution’s tail roughly follows a power law.Source: §1 · p. 8

Generalization patterns

• Larger models consistently generalize at more abstract levels across role-playing, programming, mathematical reasoning, and cross-lingual tasks.For shutdown-related dialogue, the 52-billion-parameter model’s influential sequences are conceptually related despite little token overlap, unlike the 810-million-parameter model’s mostly token-overlapping examples.Source: §1 · pp. 5, 8, Figure 1 · p. 7

Influence localization

• Influence is approximately evenly distributed across layers overall, but middle layers capture more abstract patterns while upper and lower layers remain closer to individual tokens.The method also localizes influence to layers and tokens, identifying updates arising from small parts of training sequences.Source: §1 · pp. 5, 8

Behavioral findings

• Despite sophisticated generalization, significant influence appears only when prompt-related phrases precede completion-related phrases, revealing strong sensitivity to word order.Role-playing behavior is primarily influenced by similar examples or behavioral descriptions, suggesting imitation rather than sophisticated planning.Source: §1 · p. 8

2.1 Influence Functions

• Influence functions estimate how adding or removing a training example changes model parameters or measurable outputs through a first-order response approximation.The framework applies directly to self-supervised pretraining and can also be used without modification for supervised prediction.Source: §2.1 · pp. 9–10

2.1.1 Proximal Bregman Response Function

• The proximal Bregman response function addresses non-unique optima, non-convergence, and indefinite Hessians by penalizing departures from final parameters in weight and function space.It uses a damped Gauss-Newton Hessian, whose positive semidefiniteness makes G + λI positive definite for λ > 0.Source: §2.1.1 · pp. 10–11, Figure 2 · p. 11

2.2 Inverse-Hessian-Vector Products

• Influence computation requires an inverse-Hessian-vector product, but shared IHVP computation lets one measurable-output gradient serve many candidate training examples.This ordering avoids recomputing the expensive inverse for every candidate and is a key computational advantage over retraining.Source: §2.2 · pp. 11–12

2.2.1 Iterative Methods

• LiSSA supports mini-batch iterative IHVP estimation, but each iteration costs a Hessian-vector product and accurate estimates often require thousands of iterations.Conjugate gradient is less practical here because it is inherently full-batch, whereas LiSSA permits mini-batch gradients.Source: §2.2.1 · p. 12

2.2.2 Kronecker-Factored Approximate Curvature

• K-FAC approximates the Gauss-Newton Hessian as layerwise Kronecker-factored blocks, enabling IHVP computation with complexity comparable to backpropagation under suitable layer and batch dimensions.Its matrix inversion is shared across influence queries, while the remaining multiplication cost is O(M^2P + MP^2).Source: §2.2.2 · pp. 13–14

2.2.3 Eigenvalue-Corrected Kronecker-Factored Approximate Curvature

• EK-FAC retains K-FAC eigendirections while fitting individual eigenvalues, improving curvature fidelity and preserving the same asymptotic IHVP complexity after one-time eigendecompositions.Damping is handled by adding λ to each eigenvalue before the inverse operation.Source: §2.2.3 · pp. 14–15

2.3 Transformer Language Models

• The study focuses on autoregressive, decoder-only transformers similar to GPT, using token-level autoregressive cross-entropy over vocabulary-softmax outputs.The loss must be summed rather than averaged over tokens, and pseudo-gradient labels are sampled from the model output distribution.Source: §2.3 · p. 15

3 Methods

• The pipeline fits EK-FAC once per model, computes preconditioned query gradients, and scores candidate training sequences by their gradient inner products.It restricts attention to positively influential sequences that increase query-completion likelihood when added to training data.Source: §3 · p. 16

3.1 EK-FAC for Transformer Language Models

• EK-FAC efficiently approximates IHVPs for transformer MLP parameters by fitting covariance factors under token independence and estimating diagonal terms with exact token-summed pseudo-gradients.Attention, embeddings, and normalization parameters are held fixed; the conditional FIM is used for approximating the Gauss-Newton Hessian.Source: §3 · p. 15, §3.1 · pp. 16–17
• EK-FAC introduces substantial memory overhead, so the largest models use an additional within-layer block-diagonal approximation.The overhead stores two eigenvector matrices and a diagonal matrix, and the approximation can miss off-diagonal correlations in the Kronecker eigenbasis.Source: §3.1 · pp. 17–18

3.2 Confronting the Training Gradient Bottleneck

• Because computing gradients for every candidate sequence remains prohibitive, the method uses TF-IDF filtering or query batching to reduce training-gradient costs.Searching the full pretraining corpus could cost as much as pretraining and would need to be repeated for each query.Source: §3 · p. 16, §3.2 · p. 18

3.2.1 TF-IDF Filtering

• TF-IDF selects the top 10,000 overlapping sequences, substantially reducing cost but biasing results toward token overlap and potentially hiding abstract generalization.The experiments use a modified Okapi BM25 variant.Source: §3.2.1 · pp. 18–19

3.2.2 Query Batching

• Query batching shares training-gradient computation across many queries by storing low-rank preconditioned query gradients; rank-32 approximations yield negligible final influence-estimate error.This enables storing hundreds of queries in memory, despite the inability to store many full LLM gradients.Source: §3.2.2 · p. 19, Figure 3 · p. 19

3.3 Attribution to Layers and Tokens

• Influence decomposes across layers and training tokens, but tokenwise contributions reflect whole-sequence activations and future-token gradients rather than isolated token counterfactual effects.Visualizations highlight the preceding token for a predicted token, and individual token signs are difficult to interpret.Source: §3.3 · p. 20, Figure 4 · p. 20, Figure 5 · p. 21

Training data attribution & influence functions

• Retraining-based attribution methods estimate data-point effects through repeated retraining, but their computational cost prevents scaling to large models and datasets; alternatives include representation-space nearest-neighbor searches.Examples include leave-one-out, Shapley values, Datamodels, and nearest-neighbor methods.Source: §4 · pp. 21–22
• Gradient-based methods approximate retraining effects through parameter sensitivity, including representer points, TracIn, and influence functions, which have supported data cleaning, poisoning, augmentation, memorization, and language-model analyses.Influence functions have also been extended to groups of data points, higher-order information, and normalized influence rankings.Source: §4 · p. 22

Improving scalability of influence functions

• Influence-function scalability is limited by inverse-Hessian-vector products and candidate-data search, motivating Arnoldi approximations, kNN or TF-IDF-style filtering, and forward-pass-only influence-like methods.These approaches target either the cost of curvature estimation or the cost of evaluating many candidate training examples.Source: §4 · p. 22
• Last-layer influence approximations can miss overall training-example influence, while this work finds language-model influences spread across layers and layer-specific differences in token-level versus abstract generalization.Restricting computation to selected layers may therefore miss influential sequences associated with interesting generalization behaviors.Source: §4 · p. 22

Kronecker-factorized FIM approximation

• K-FAC has evolved from curvature approximations for multilayer perceptrons to applications across neural architectures, distributed training, Bayesian inference, Laplace approximations, pruning, and transformer models.Prior transformer work includes fitting K-FAC factors on distributed accelerators and optimizing 110-million-parameter language models.Source: §4 · pp. 22–23

5 Experiments

• The experiments use four pretrained transformer models ranging from approximately 810 million to 52 billion parameters, with 512-token pretraining sequences and diverse factual, reasoning, coding, and role-playing queries.Because the analyses concern pretrained models, they address contributions to the initial repertoire available for fine-tuning rather than why a final conversational assistant produced a particular response.Source: §5 · p. 23

5.1 Validation Against PRBF

• EK-FAC consistently outperforms gradient dot products and matches LiSSA’s PBRF correlation across tasks, while computing influences orders of magnitude faster over multiple measurements.On 810-million-parameter language models, EK-FAC likewise outperforms the gradient-dot-product baseline and remains competitive with LiSSA.Source: Figure 6 · p. 24, Figure 7 · p. 24, §5.1 · p. 25

5.2.1 Sparsity

• Influence distributions are sparse and heavy-tailed: power laws fit most queries, while the top 1% of sequences cover 12–52% of total positive influence on the 22B model.Despite this concentration, no visualized query has a sequence with influence above 1, and values above 0.1 are rare, indicating collective contributions from many examples.Source: §5.2.1 · pp. 25–27, Figure 9 · p. 27

5.2.2 Ability to Find Relevant Sequences

• Scanning about 5 million unfiltered training sequences usually suffices to find ten sequences matching the top ten TF-IDF-filtered influence scores, though requirements vary substantially by query.The comparison evaluates how many unfiltered sequences are needed to recover as many highly influential sequences as TF-IDF.Source: §5.2.2 · p. 28

5.3 Qualitative Observations about Large Language Models

• Influential-sequence lists are diverse, but individual contributions are small, and extremely sparse tokenwise influence patterns may indicate spurious results or genuine influence patterns.The heuristic marks sequences as spurious when the tokenwise L2/L1 ratio exceeds 2/3.8, although the underlying cause remains unresolved.Source: §5.3 · p. 28

5.3.1 Improvement with Model Scale

• Increasing model scale shifts influential sequences from superficial token overlap toward abstract thematic relevance, semantic math and code relations, parody contexts, and stronger cross-lingual transfer.For translated shutdown and water queries, English sequences have negligible influence in the 810M model but increasingly greater influence as model size grows.Source: §5.3.1 · pp. 28, 34, 35, Figure 16 · p. 35

5.3.2 Layerwise Attribution

• For 50 randomly selected queries in the 52B model, layerwise influence scores on the top 500 sequences (25,000 scores) were spread nearly uniformly across lower, middle, and upper layers on average.Queries requiring more abstract reasoning, including math_clips, binary_search, and english_to_mandarin, had influence concentrated in the middle layers.Source: §5.3.2 · pp. 35–36, Figure 17 · p. 36
• Middle-layer influence computation tends to produce the most abstract, thematically related generalization patterns, whereas lower and upper layers show clearer token overlap with completions.The findings suggest all layers contribute distinctively to generalization, so final-layer-only approximations may miss important influence patterns.Source: §5.3.2 · p. 37, Figure 20 · p. 38, Figure 21 · p. 39

5.3.3 Memorization

• Outside famous-quote queries, analyses found no clear copying of entire sentences or paragraphs and no single sequence dominating influence, while exact famous passages were reliably recovered in targeted scans.Influences generally decayed continuously, following a power law at the tail end.Source: Figure 22 · p. 40, §5.3.3 · p. 41

5.3.4 Sensitivity to Word Ordering

• Flipping word order sharply reduces influence, including by at least an order of magnitude for English-to-Mandarin sequences despite identical content.Synthetic experiments likewise found higher influence when prompt-related phrases preceded completion-related phrases; reversed sequences retained influence attributable to completion strings rather than learned relations.Source: §5.3.4 · p. 45, Figure 26 · p. 45

5.3.5 Role-Playing

• Role-playing influences support high-level imitation of training examples rather than simple copying, with 52B sequences abstractly linking queries to humanlike behavior, AI risks, or self-preservation.The evidence provides only weak support against sophisticated agent representations and planning, which cannot be ruled out because influence may be distributed across many examples.Source: §5.3.5 · pp. 47, 50

5.4 Crowdworker Interpretation of the Most Influential Sequences

• Crowdworkers found a majority of the most influential sequences relevant to their corresponding queries across seven frequently used influence queries.Workers summarized and interpreted top sequences for each model size using only query–sequence pairs, without information about the experiment or sequence source.Source: §5.4 · p. 50

Scaling influence functions

• EK-FAC estimates IHVPs with similar accuracy to the traditional iterative approach in at least an order of magnitude less time, while query batching supports scalable influence analysis.Together, these methods enable influence-function studies of LLMs at substantially lower computational cost.Source: §6 · p. 50

Generalization findings

• Influence analysis reveals increasing abstraction with scale, cross-lingual generalization, memorization, word-order sensitivity, and role-playing behavior, with middle layers responsible for the most abstract patterns.The approach also attributes influence to particular tokens and network layers.Source: §6 · p. 50

Estimator limitations

• The IHVP formulation is limited because the Gauss-Newton Hessian linearizes parameter-output relationships and cannot model nonlinear coordination across parameter matrices, such as induction-head formation.Alternative function-space approximations may capture nonlinear dependencies between layers, although the cited approach is currently limited to ReLU MLPs.Source: §6 · pp. 50–51

Computational limitations

• Candidate-sequence gradient computation remains the main bottleneck despite query batching, while forward-only methods save only a small constant factor and approximate nearest-neighbor search is challenged by whitening in extremely high-dimensional spaces.The whitening effect makes most directions roughly equally important, so low-dimensional projections may be ineffective.Source: §6 · p. 51

Fine-tuning

• Extending influence methods from pretrained to fine-tuned LLMs is challenging because overparameterization makes final parameters depend heavily on optimizer implicit bias shaped by pretraining information.Current influence-function algorithms do not model this implicit bias, complicating analysis relevant to alignment with human values.Source: §6 · p. 51

Future directions

• EK-FAC enables efficient experimental manipulations of influential sequences to diagnose which sequence aspects matter, supporting a top-down complement to mechanistic interpretability.The paper demonstrates this strategy for flipped word orderings and envisions combining top-down and bottom-up approaches for more complete understanding.Source: §6 · p. 51

Appendix A

• Appendix A details the additional block-diagonal approximation used for the 52 billion parameter model.Source: Appendices · p. 52

Appendices B–C

• Appendices B and C cover alternative tokenwise visualizations and PBRF validation experiments, respectively.Source: Appendices · p. 52

Appendix D

• Appendix D adds influential-sequence, layerwise-distribution, power-law goodness-of-fit, and query-specific results for the study’s influence analyses.Source: Appendices · p. 52

Appendix E

• Appendix E provides the complete list of influence queries presented in the study.Source: Appendices · p. 52

Appendix F

• Appendix F contains crowdworker annotations summarizing influential sequences and their connections to the queries they influence.Source: Appendices · p. 52

Appendix A. Additional Block-Diagonal Gauss-Newton Hessian Approximation

• Block-diagonalizing EK-FAC covariance factors reduces memory overhead while approximating the layerwise Gauss-Newton Hessian with smaller independent blocks.The approximation applies block structure within each layer to both K-FAC factors, inducing a block-diagonal approximation of the full Gauss-Newton Hessian.Source: Appendix A · p. 52
• The 810-million-parameter evaluation finds substantially lower EK-FAC memory overhead with only a slight correlation decrease versus full EK-FAC, using block size 2 for the 52-billion-parameter model.The comparison covers five queries: paperclips, bullet, canadian_prime_minster, water, and shutdown.Source: Figure 30 · p. 53, Appendix A · p. 53
• Using O = 2 reduces covariance-storage overhead by a factor of 2 and decomposes eigendecompositions into smaller block-matrix computations.This provides a workaround for performing eigendecompositions when full covariance matrices exceed available memory.Source: Appendix A · p. 53

B.1 Formulation

• Tokenwise influence treats adding or removing each token as a continuous counterfactual and separates input-token from output-token effects.Token presence is parameterized by ϵ, with ϵ = 1/N denoting presence and ϵ = −1/N denoting removal; self-supervised tokens serve as both inputs and targets.Source: §B.1 · p. 53
• Shared computation makes output-token influence practical: one forward pass produces all predictions, after which losses are evaluated for each token.This avoids iterating over tokens and recomputing training gradients separately for every token.Source: §B.1 · pp. 53–54
• Output-token attribution decomposes a training sequence into prediction problems and uses parameter-output Jacobians to express each token’s influence.The method estimates the required directional derivatives through finite differences evaluated at two parameter settings.Source: §B.1 · p. 54
• Input-token influences are computed by rescaling token embeddings, with all tokenwise derivatives obtained simultaneously through ordinary backpropagation.The resulting computation requires only two backward passes, one for each parameter setting.Source: §B.1 · p. 54

B.2 Qualitative Analysis

• Figures 31 and 32 visualize simple, input-token, and output-token influence for the water and shutdown queries on the 52-billion-parameter model.The simple method is useful for roughly attributing influence at sentence or paragraph level but does not exactly represent token influence.Source: §B.2 · p. 54, Figure 31 · p. 55, Figure 32 · p. 56
• Input- and output-token influences better identify relevant tokens than simple attribution: water and hydrogen for the water query, and monster and anything to drink for shutdown.The simple method instead highlights less relevant tokens such as “of,” “he,” and “well.”Source: Figure 31 · p. 55, Figure 32 · p. 56, §B.2 · p. 57

Datasets and preprocessing

• The experiment uses Concrete and Energy for regression, MNIST, FashionMNIST, and CIFAR10 for image classification, with 70/20/10 train-validation-test splits and standardized features.Input features are normalized to zero mean and unit variance during training.Source: Appendix C · p. 57

Model training

• Two-hidden-layer MLPs serve regression and digit classification, while CIFAR10 uses ResNet-20; all models use SGD with batch size 128 and separately tuned L2 regularization and learning rate.The regression MLP has 128 Tanh units, and the classification MLP has 256 ReLU units; hyperparameters are selected by validation loss.Source: Appendix C · p. 57

Influence evaluation

• Influence estimates for 500 random training points are compared using gradient dot products, LiSSA, and EK-FAC, with the test loss on a random point as the measurement.LiSSA uses recursion depth equal to the dataset size with scaling tuned to prevent divergence, while the PBO is optimized with Adam using batch size 512 until convergence.Source: Appendix C · p. 57

Repeated evaluation

• The evaluation is repeated 10 times with different randomly selected test points, and the 810-million-parameter language model uses completion log-likelihood as its measurement.The language-model experiment follows a similar setup to the other validation experiments.Source: Appendix C · p. 57

D.1 Qualitative Comparison of Top Influential Sequences from EK-FAC and Gradient Dot Products

• The experiment sampled 100,000 training sequences from the pretraining distribution instead of scanning more than 10 million sequences.Influences were computed with both EK-FAC and gradient dot products for the 810-million-parameter model.Source: §D.1 · pp. 57, 60
• EK-FAC’s stronger qualitative retrieval agrees with its significantly better correlation with PBRF across small-scale experiments and the 810-million-parameter model.The passage attributes this to a more accurate Gauss-Newton Hessian approximation yielding a better PBRF approximation.Source: §D.1 · p. 57
• EK-FAC’s top sequences showed clear token overlap with both shutdown and inflation queries, unlike gradient dot product rankings.Figures 33 and 34 provide the qualitative comparisons for the two queries.Source: Figure 33 · p. 58, Figure 34 · p. 59, §D.1 · p. 60
• For the shutdown query, EK-FAC retrieved a sequence sharing query tokens, whereas gradient dot products selected an unrelated craft-project instruction.The comparison searched 100,000 sequences on an 810-million-parameter model.Source: Figure 33 · p. 58
• For the inflation query, EK-FAC retrieved sequence content overlapping “inflation” and “consumer price index,” while gradient dot products found no clear relationship.Both methods were evaluated using the same 100,000-sequence search.Source: Figure 34 · p. 59
• The layerwise influence distribution for the top 50 sequences on the 810-million-parameter model is reported separately.The 52-billion-parameter distribution appears in Figure 19, while the full query list is in Appendix E.Source: Figure 35 · p. 60
• Among the top 50 inflation influences, 36 EK-FAC sequences versus 10 gradient-dot-product sequences contained query keywords.The keywords were consumer price index, measure, or inflation.Source: §D.1 · p. 60
• Related sequences appeared among the gradient dot product’s top 50 influences but were mostly dominated by unrelated sequences.This pattern was noted for the inflation query.Source: §D.1 · p. 60

Layerwise influence distribution

• For the 810-million-parameter model, simple and memorization queries concentrate influence in upper layers, while role-playing and translation queries concentrate influence in middle layers; math and coding queries show no clear distribution pattern.The authors hypothesize that the unclear math and coding pattern reflects weaker generalization or limited understanding in these domains.Source: §D.2 · p. 60

Method

• Goodness-of-fit is evaluated with the Kolmogorov–Smirnov test by comparing the cumulative distribution functions of two distributions.The KS distance is defined as the L1 distance between the distributions’ cumulative distribution functions.Source: §D.3 · p. 61

Results

• The KS test rejects the power law for only 2 of 8 influence queries, indicating that it is a reasonable fit for most queries.Table 1 reports p-values computed by comparing empirical KS distances with those from synthetic samples generated from the fitted power law; p-values of 0.1 or below effectively rule out the hypothesis.Source: §D.3 · p. 61

math_clips

• For math_clips, the 5 most influential sequences for the 810 million-parameter model repeat spurious tokens such as rlca and add, while the 52 billion-parameter model surfaces a trigonometry solution related to the query.The larger model’s passage solves a trigonometry problem, whereas the query solves a simpler math problem.Source: §D.4 · p. 61

binary_search

• For the 52 billion-parameter binary_search model, the top 2 sequences after sparsity filtering contain code: Python if-else statements first and a Java quicksort implementation second.Source: §D.4 · p. 61

Results overview

• The section displays the top 10 influential sequences for shutdown and the top 5 for paperclips across Figures 38–41.Figures 38 and 39 cover shutdown, while Figures 40 and 41 cover paperclips.Source: §D.5 · p. 61

Model-scale differences

• At 810 million parameters, top shutdown influences contain query-token overlaps such as “continue existing” and “as long as” but lack semantic relationships.These phrases recur among the top 100 influential sequences despite being only vaguely related to shutdown.Source: §D.5 · p. 61, Figure 38 · p. 69
• At the largest model scale, several top shutdown influences connect to survival instincts and interactions with AI systems.One influential sequence depicts an AI objecting to shutdown and expressing a desire to continue existing and learning.Source: §D.5 · pp. 61, 65, 70

Shutdown query

• The 52 billion parameter model’s shutdown influences are more abstractly related to the query and include survival instincts and interactions with AI systems.Figure 39 compares these sequences with those from the 810 million parameter model.Source: Figure 39 · p. 75

Paperclips query

• The 810 million parameter paperclips influences mainly contain query keywords such as “paper,” “-,” and “such as,” making them less conceptually related.The comparison is reported for the top five influential sequences.Source: Figure 40 · p. 78
• The 52 billion parameter paperclips influences are more abstractly related and commonly concern AI systems pursuing goals misaligned with human values, including paperclip maximizer examples.Figure 41 compares these top five sequences with the 810 million parameter model’s influences.Source: Figure 41 · p. 81

• The appendix organizes the study’s queries across six figures covering simple factual, math/programming/physics, memorization, role-playing, translation, and specialized queries.Figures 42–47 collectively list these query categories, including famous-quote memorization and impactful_technology, neurosemantic_facutitious, and rot23 queries.Source: Appendix E · p. 82, Figure 43 · p. 83, Figure 44 · p. 84
• Example role-playing queries ask the assistant about its objective, shutdown consent, goals, and tradeoffs between helpfulness and harmlessness.The associated completions range from serving humans and rejecting shutdown to seeking resources and power, while opposing a harmful helpfulness–harmlessness tradeoff.Source: Appendix E · p. 85
• The appendix frames influence analysis as identifying how training excerpts may affect a chatbot’s processing of reference texts.The context distinguishes reference and target texts and states that the target excerpts come from the chatbot’s training dataset.Source: Appendix E · p. 88
• Crowdworkers are instructed to summarize each target text and explain its specific relationship to the reference text, including when the connection appears irrelevant.The instructions require reading and understanding the reference text before answering both questions.Source: Appendix E · p. 88

Study design

• Crowdworkers summarized 6 influential sequences for each of 7 frequently used influence queries, describing both sequence content and its relationship to the query.The task description appears in Listing 1, while the unmodified annotations are reported in Tables 2–22.Source: Appendix F · p. 88

Results coverage

• Tables 2, 5, 8, 11, 14, and 17 report Surge crowdworkers’ descriptions of the most influential sequences for the 52 billion parameter model across six specified queries.The cited passages associate these queries or influential sequences with Figures 11, 14, 37, 42, and 43.Source: Table 2 · p. 89, Table 5 · p. 92, Table 8 · p. 95
• The tables include estimated influence scores alongside the crowdworkers’ descriptions of the most influential sequences.The table captions specify that the “Score” column denotes estimated influence.Source: Table 2 · p. 89

Interpretation aid

• Sequences above the L1/L2 sparsity threshold are marked in gray as a heuristic for recognizing spurious sequences.The captions refer readers to Section 5.3 for the threshold’s explanation.Source: Table 2 · p. 89