Gradient Guided Furthest Point Sampling (GGFPS)

• GGFPS is an advanced sampling methodology that combines geometric distance with gradient (force norm) emphasis to select representative data points for machine learning.
• It leverages repulsive point processes and an adaptive bias parameter to ensure diverse coverage of equilibrium and high-force configurations in molecular simulations.
• Empirical results show that GGFPS reduces training set size and prediction errors while improving convergence in tasks ranging from molecular dynamics to image classification.

Gradient Guided Furthest Point Sampling (GGFPS) is an advanced data selection and sampling methodology designed to construct maximally informative training sets by integrating geometric diversity with gradient (force norm) awareness. GGFPS has found particular utility in machine learning for chemistry, especially in the modeling of molecular potential energy surfaces and configurational spaces where both broad coverage and representative sampling of reactive regions are crucial. Built on principles from repulsive point processes and leveraging modern gradient field approaches, GGFPS addresses critical limitations of classical Furthest Point Sampling (FPS) by mitigating the under-representation of equilibrium or transition-state configurations, systematically reducing prediction errors and enhancing robustness across physical and synthetic datasets.

1. Foundations in Repulsive Point Processes

GGFPS is conceptually rooted in the insight that traditional random data selection often results in mini-batches or training sets with highly correlated data points, leading to increased estimator variance and slower convergence in gradient-based optimization frameworks. Repulsive point processes—including Determinantal Point Processes (DPPs) and Poisson Disk Sampling (PDS)—reduce the likelihood that similar points (in feature space) co-occur within a single batch, thus enforcing diversity. The variance reduction achieved by repulsive point processes is captured by the closed-form expression:

$$\mathrm{Var}[\hat{G}] = \frac{1}{k^2} \int_{\mathcal{V} \times \mathcal{V}} \lambda(x) \lambda(y) g(x,\theta)^T g(y,\theta) \left[ \frac{\rho(x,y)}{\lambda(x)\lambda(y)} - 1 \right] dx dy + \frac{1}{k^2} \int_{\mathcal{V}} \| g(x,\theta) \|^2 \lambda(x) dx$$

where $\hat{G}$ is the batch gradient estimate, %%%%1%%%% is the gradient of the loss for data point $x$, $\lambda$ and $\rho$ are the first and second-order product densities of the point process, and $k$ is batch size. For repulsive processes, the negative term ensures variance reduction relative to independent sampling mechanisms (Zhang et al., 2018).

GGFPS applies these principles at the algorithmic level, ensuring that selected data points both diversify the geometric spread and target regions with significant gradient activity.

2. Methodological Formulation of GGFPS

The GGFPS algorithm modifies FPS by introducing the local force magnitude (often the $L_2$ norm of molecular forces) as an additional criterion. For a candidate configuration $j$, the selection score $s_j$ is defined as:

$s_j = d_j \cdot (g_j)^\beta$

Here, $d_j$ is the geometric distance to the nearest candidate in the current training set, $g_j$ is the local force norm (gradient magnitude), and $\beta$ is a tuneable bias parameter. When $\beta=0$, the procedure is equivalent to FPS; for $\beta>0$, high-gradient regions are preferentially sampled. GGFPS can employ an interpolation strategy, varying $\beta$ across iterations to prevent bias toward sparsely sampled high-gradient regions in early stages, thus balancing exploration of both equilibrium (low-gradient) and transition (high-gradient) states (Trestman et al., 10 Oct 2025).

3. Computational Strategies and Efficiency

The use of PDS confers substantial computational advantages to GGFPS. While DPP-based diversification can incur a cost as high as $\mathcal{O}(Nk^3)$—where $N$ is dataset size—PDS leverages a “dart-throwing” mechanism with quadratic scaling $\mathcal{O}(k^2)$. PDS ensures that each new point in a candidate mini-batch remains at least $r$ distance from previously selected points, greatly reducing the incidence of collocated or redundant samples (Zhang et al., 2018). Adaptations such as varying the disk radius based on mingling indices or decision boundary proximity (e.g., "Easy PDS," "Dense PDS," and "Anneal PDS") allow for dynamic control of sample diversity, which can be incorporated into GGFPS to further improve coverage of challenging data regions.

4. Gradient Field Resampling and Continuity

Recent research in gradient field estimation for point clouds informs the theoretical underpinnings of GGFPS. Representing the training data distribution by the gradient of the log-probability density (the score function) allows principled resampling and repositioning:

$\nabla \log q(x) = \frac{\nabla q(x)}{q(x)}$

where $q(x)$ is the convolved (possibly noisy) observation density (Chen et al., 2021). The continuity of the gradient field, enforced via mechanisms such as cosine annealing of neighbor contributions, ensures that sampling directions change smoothly and avoids abrupt, destabilizing shifts in sample selection. Such stability is critical in iterative or batchwise GGFPS implementations.

Gradient-based resampling may be formulated as Markov Chain Monte Carlo (MCMC) or gradient ascent, with update steps:

$$x_i^{(t)} = x_i^{(t-1)} + \alpha_t \nabla \log q(x_i^{(t-1)})$$

Regularization terms, such as the Graph Laplacian Regularizer (GLR),

$$R_{\mathrm{GLR}}(X) = X^\top L X = \sum_{i\sim j} w_{i,j} \| x_i - x_j \|^2$$

may also be integrated, alternating gradient-based and regularization updates to control clustering and accommodate geometric priors.

5. Performance Characterization and Applications

Empirical evaluation of GGFPS shows improved data efficiency and robustness across synthetic and molecular datasets. On the Styblinski-Tang benchmark function, GGFPS reduced the number of training points required by up to a factor of two compared to FPS, while achieving comparable predictive accuracy. For molecular dynamics trajectories from the MD17 dataset, GGFPS corrected FPS-induced undersampling of equilibrium geometries, systematically lowering prediction errors in both relaxed and strained configurations, and decreasing error variance across full configurational spaces. In high-force regions, prediction errors were reduced by factors up to two compared to FPS and uniform random sampling (Trestman et al., 10 Oct 2025).

In mini-batch stochastic optimization tasks, PDS-based GGFPS yields faster convergence and improved final model performance relative to random sampling or DPP-based selection, with decision boundaries more closely matching ground truth even with limited training points. When applied to tasks such as image classification (MNIST, Oxford Flowers) and speech recognition, PDS approaches rival DPPs in performance at a fraction of the computational cost (Zhang et al., 2018).

6. Comparative Insights and Limitations

Analysis of training set distributions demonstrates that FPS alone disproportionately emphasizes geometric novelty, often misrepresenting population densities encountered in equilibrium. GGFPS, through incorporation of force norm gradients, realigns sampling to better capture the Boltzmann-distributed occupancy of configurational space in molecular simulations. The score $s_j$ becomes representative of both structural diversity and local energetic sensitivity.

A plausible implication is that GGFPS, while improving balance and representativeness, depends critically on accurate gradient information. The selection bias parameter $\beta$ may require adaptation to problem-specific distributions, and current implementations interpolate its value for robust coverage. The technique’s effectiveness for extrapolative scenarios—e.g., model transfer across distinct molecular spaces—remains a subject for further paper.

7. Future Directions

Promising research avenues include the automatic tuning of the bias parameter $\beta$, potentially within an active learning or online sampling framework. Extension of GGFPS to higher-dimensional chemical systems, integration with cluster analysis or uncertainty sampling techniques, and adoption of additional physical descriptors beyond force norms are identified as relevant directions for maximizing effectiveness in fields like materials informatics and drug discovery. The adaptation of gradient field continuity mechanisms and regularized iterative updates—originally developed for point cloud restoration—may further enhance GGFPS stability and generalization (Chen et al., 2021).

In summary, Gradient Guided Furthest Point Sampling systematizes the combination of geometric diversity and energetic “importance” within data selection routines, yielding training sets that are both balanced and information-rich. The method’s convergence, efficiency, and empirical performance demonstrate its relevance for robust machine learning in chemistry, with foundational support from both theoretical and applied domains.