Functional Group Representation (FGR)

• FGR is a representation method that encodes molecules and functional data by grouping chemically or statistically similar variables into interpretable units.
• It uses both curated and data‐mined substructures to generate compact latent embeddings that enhance prediction accuracy and enable clear feature attribution.
• FGR methodologies improve model robustness in molecular and functional analyses, yielding state-of-the-art performance on diverse benchmark datasets.

A Functional Group Representation (FGR) encodes information about a system—typically in chemistry, molecular science, or functional data analysis—by leveraging domain-specific groupings of variables or substructures. In molecular machine learning, FGR denotes explicit representations of molecules in terms of functional groups (FGs)—distinct chemical substructures with recurring reactivity patterns—rather than conventional atom-based encodings. In functional data analysis, FGR formalizes predictors and coefficients as collections (“groups”) of functions, often facilitating group-sparse regression or functional variable selection. Across its domains, FGR enables interpretability, inductive bias toward chemically or functionally meaningful motifs, and data-efficient statistical modeling.

1. Formal Frameworks of FGR

Molecular Representation

FGR in molecular learning comprises two principal components:

• Curated Functional Groups (FG): Predefined substructure patterns specified via SMARTS strings, reflecting core chemical knowledge. For a molecule with SMILES representation $S$, let $\mathcal{FG} = \{FG_1, \ldots, FG_a\}$ with $x_{FG}^{(i)} = 1$ iff $FG_i$ matches $S$.
• Mined Functional Groups (MFG): Data-driven substructures obtained by sequential pattern mining over large chemical corpora. $\mathcal{MFG} = \{MFG_1, \ldots, MFG_b\}$ with $x_{MFG}^{(j)} = 1$ iff $MFG_j \subset S$.

The combined molecular input $x_G = x_{FG} \oplus x_{MFG} \in \{0,1\}^{a+b}$ is mapped to a latent embedding $z_G = W_e x_G + b_e$ via a feedforward encoder, supporting downstream property prediction and unsupervised pretraining (Balaji et al., 11 Sep 2025).

Rule-based FG tokenization at the level of the molecule—assigning each atom to a unique FG and segmenting canonical SMILES into a sequence of FG-specific tokens—underlies the FG-aware representation in models such as FARM (Nguyen et al., 2024) and Group SELFIES (Cheng et al., 2022), where group tokens package entire substructures as primitive units in the molecular string, preserving chemical robustness.

Functional Data Analysis

In high-dimensional functional regression, FGR identifies each group as a single functional predictor $\mathcal{FG} = \{FG_1, \ldots, FG_a\}$0 (with $\mathcal{FG} = \{FG_1, \ldots, FG_a\}$1 a separable Hilbert space, e.g., $\mathcal{FG} = \{FG_1, \ldots, FG_a\}$2). A sample comprises $\mathcal{FG} = \{FG_1, \ldots, FG_a\}$3, with scalar response $\mathcal{FG} = \{FG_1, \ldots, FG_a\}$4. The regression functional parameterizes groupwise effects as $\mathcal{FG} = \{FG_1, \ldots, FG_a\}$5, reflecting a group structure over predictors, and penalizes solutions via groupwise Hilbert-norm penalties. Each group thus denotes all components and their associated coefficient functions indexed by a single $\mathcal{FG} = \{FG_1, \ldots, FG_a\}$6 (Mahzarnia et al., 2021).

2. Implementation Methodologies

Preprocessing and Vocabulary Construction

• In FGR for molecules, generative mining of MFG tokens uses frequent pattern mining on large SMILES datasets via iterative pair-token merges, expanding a data-driven fragment vocabulary. SMARTS-based matching exhaustively labels curated FGs per molecule.
• FARM assigns atoms to one of 101 predefined non-ring FGs or corresponding ring types, segments SMILES into contiguous FG runs, and collapses each into FG-specific tokens, expanding the lexicon from $\mathcal{FG} = \{FG_1, \ldots, FG_a\}$793 to $\mathcal{FG} = \{FG_1, \ldots, FG_a\}$814,000 tokens (Nguyen et al., 2024).
• Group SELFIES encodes molecules using user-specified group sets $\mathcal{FG} = \{FG_1, \ldots, FG_a\}$9, matching subgraphs greedily and substituting substructure tokens with explicit attachment indices (Cheng et al., 2022).

Encoder Architectures

• Feedforward encoders and autoencoders on $x_{FG}^{(i)} = 1$0 input achieve compact and decorrelated latent representations, leveraging both curated and mined FGs (Balaji et al., 11 Sep 2025).
• FG-aware LLMs operate on FG-tokenized SMILES; GNNs process the FG-graph (nodes: FGs, edges: bonds between FGs), optionally combining with knowledge-graph derived embeddings (Nguyen et al., 2024).
• Dual-view contrastive learning, aligning atom-level and FG-graph representations, enforces unified chemical semantics (Nguyen et al., 2024).

Functional Regression Optimization

• Basis expansion of functional coefficients: $x_{FG}^{(i)} = 1$1, with norm regularization $x_{FG}^{(i)} = 1$2 leads to group lasso-like objectives for latent parameter selection (Mahzarnia et al., 2021).
• ADMM solvers with groupwise soft-thresholding yield globally optimal, group-sparse solutions for the infinite-dimensional functional regression problem.

3. Theoretical Guarantees and Model Properties

• Groupwise Sparsity: Imposing $x_{FG}^{(i)} = 1$3-norm or Hilbert-norm penalties across groups enforces entire-zero solutions at the group level, yielding interpretable subset selection in both functional regression and molecular modeling (Mahzarnia et al., 2021, Balaji et al., 11 Sep 2025).
• Robustness: Group SELFIES offers valency-tracked guarantees, ensuring every group-tokenized string decodes to a chemically valid molecular graph—any excessive bonding request is ignored or downgraded, preserving chemical realism (Cheng et al., 2022).
• Inductive Bias: Packaging frequent and chemically meaningful substructures as primitive tokens introduces strong inductive bias toward plausible motifs and scaffold preservation, enhancing data efficiency and generation quality (Cheng et al., 2022).

4. Practical Applications and Benchmarks

Molecular Machine Learning

FGR-based frameworks achieve state-of-the-art results for molecular property prediction across 33 benchmark datasets encompassing physiology, biophysics, quantum mechanics, bioactivity, pharmacokinetics, and more. Average ROC-AUC and RMSE improvements of 1.5%–8.7% over previous best models are reported for standard datasets such as BBBP, Tox21, SIDER, QM7–QM9, and others (Balaji et al., 11 Sep 2025, Nguyen et al., 2024). Crucially, FGR enables feature attribution directly on interpretable FGs, mapping predictive relevance at the physicochemical motif level.

Functional Data Analysis

Functional group sparse regression (e.g., MFG-LASSO) on fMRI data (p=116 brain ROIs, n=290 subjects) for intelligence and ADHD phenotypes selects neurologically coherent regions, with prediction RMSE markedly reduced compared to functional OLS or ridge regression (Mahzarnia et al., 2021). Simulation benchmarks demonstrate perfect or near-perfect selection of active groups and oracle-risk RMSE for well-specified models.

Molecular Generation

Group SELFIES improves generative modeling: randomly sampled group-SELFIES strings more faithfully reproduce empirical distributions of synthesizability (SAScore) and drug-likeness (QED), and achieve 3.5× improvement in Fréchet ChemNet Distance (FCD) relative to atom-SELFIES VAE baselines on benchmarks such as MOSES (Cheng et al., 2022).

5. Interpretability and Attribution

FGR enables explicit linkages between predictions and specific molecular or functional motifs:

• Attribution algorithms (Integrated Gradients, GradientShap, permutation/ablation) quantify the contribution of every FG and descriptor to a model’s output (Balaji et al., 11 Sep 2025).
• High-attribution features systematically align with structure-activity relationships (e.g., chalcogens and heterocycles for BACE inhibition, amines/aromatics for BBB permeability), facilitating mechanistic hypothesis generation and guiding experimental design.
• In regression settings, the group lasso penalty ensures selected variables correspond to entire interpretable units (e.g., brain ROIs), simplifying post-hoc domain-specific analysis (Mahzarnia et al., 2021).

6. Extensions and Future Directions

Potential future developments for FGR include:

• Expansion of group vocabularies to novel and rare motifs (e.g., fused rings, macrocycles) via larger pretraining sets (Nguyen et al., 2024).
• Incorporation of geometric and 3D stereochemical descriptors within the FGR pipeline.
• Integration of FGR into generative or inverse design models, enabling the construction of novel compounds as compositions of functional group tokens.
• In functional regression, extension to vector-on-function and operator learning settings, and adaptation for multi-modal and non-Euclidean domains.

A plausible implication is that the sustained expansion and refinement of functional group vocabularies, together with attribution and robust generative algorithms, will deepen the synergy between chemical and statistical interpretability, bridging the gap between machine-learned representations and domain-theory-driven analysis.