Combining additivity and active subspaces for high-dimensional Gaussian process modeling

Gaussian processes are widely used for regression and classification, but they struggle in high-dimensional settings due to the curse of dimensionality. This paper proposes a hybrid multi-fidelity model that combines additivity and active subspaces to address these challenges in high-dimensional Gaussian process modeling.

The gist

The proposed multi-fidelity approach combines a first order additive model as a coarse level with a linear embedding model as a finer level, using an active subspace method to learn the intrinsic dimension, which is shown to improve performance over standard GPs when either additivity or active subspaces are present.

Structural Assumptions and Model Components

The paper focuses on two promising structural assumptions for high-dimensional GP modeling: additivity and linear embeddings. Additive models decompose the function into components involving only a few variables, limiting variable interactions, while linear embeddings capture complex interactions by projecting the problem onto a low-dimensional subspace. The authors detail their respective strengths and weaknesses, noting that additive models typically capture very well the main trends from high-dimensional data, but their full inference is combinatorially intractable due to interaction terms. Linear embeddings offer great scalability and allow capturing complex interactions, but only as long as the intrinsic dimension is low.

The Multi-Fidelity Strategy

To combine these structures without complexifying inference, the authors introduce a multi-fidelity approach based on an auto-regressive (AR) model. The coarse level is set to a first order additive model, and the finer level is a linear embedding model. The predictive equation for this combination is given by:

(2) YE(x) = ρYC(x) + δ(Ax), where YC(x) ⊥ δ(Ax). This structure allows the linear embedding to learn the remaining high-order interactions. The authors also discuss a recursive formulation of this AR multi-fidelity model, which provides predictive quantities at all fidelity levels.

Active Subspace Learning and Combination

For the active subspace (AS) component, the paper discusses inference of the intrinsic dimension within one- or twostage methods. They propose a two-stage approach: first fitting a high-dimensional GP to estimate the AS matrix C, and then using this matrix to learn a low-dimensional GP on projected data. Furthermore, they present an advanced method where all hyperparameters of the low-dimensional GP are learned simultaneously via the likelihood of that model. This involves calculating derivatives such as ∂ log L / ∂θi which depend on terms like ∂Ei,j / ∂θi, allowing for the learning of parameters in a single step.

Empirical Validation and Findings

The proposed multi-fidelity plus AS model (ASMF) was compared against baselines, including standard GPs (Ref), first order additive models (Add), and linearly embedded GPs (AS). The results confirm that the multi-fidelity approach improves over a standard GP when either additivity or active subspaces are present, and importantly, the performance does not degrade when such structures are absent. Specifically, the best results are obtained when both structures are present, suggesting that the dedicated GP models perform best for problems with simultaneously additive and AS structures. The budget effect is noted: more data is beneficial to all models, but the effect is most striking on AS models, suggesting a minimal amount of data is essential for robust inference. Finally, the paper concludes by highlighting that future research could explore non-linear dimension reduction or sequential procedures.

Conclusion and Perspectives

The authors propose a simple solution to combine additivity and low intrinsic dimensionality through the multi-fidelity model, which is described as simple to construct and robust to incorrect assumptions. The promising results suggest improvements in GP hyperparameter inference, the potential for combining sparse GP models, and the exploration of non-linear dimension reduction techniques. Future work could involve the alignment of these goals compared to Bayesian optimization, exploring how these aspects synergize in sequential decision-making processes.

How it works

1. The model utilizes a multi-fidelity strategy where a coarse level corresponds to a first order additive model and a high-fidelity level is a GP on an active subspace whose dimension is learned.

2. The core predictive equation for the combination is structured as: YE(x) = ρYC(x) + δ(Ax), ensuring orthogonality between the coarse and fine components.

3. For AS-based GPs, the intrinsic dimension can be inferred using one- or two-stage methods, where an AS matrix C is estimated from a high-dimensional GP to define a coordinate system for projecting data into a lower dimension.

Structural Assumptions and Model Components

The paper concentrates on two efficient structural assumptions: additivity and linear embeddings. Additive models decompose the function into components involving only subsets of variables, while linear embeddings learn the most important directions of variation by mapping the input space to a lower-dimensional space via a matrix A.

Based on a rigorous analysis of the provided scientific paper, here are specific improvements for AI systems, categorized by technical capability:

)1. Enhanced High-Dimensional Surrogate Modeling (The Core Improvement)

The primary improvement lies in developing a novel surrogate modeling framework capable of handling high-dimensional input spaces where standard Gaussian Processes (GPs) fail due to the curse of dimensionality.

• An AI system utilizing this paper can perform highly accurate regression/prediction for complex functions defined over many variables (e.g., synthetic functions, physical simulations, or large datasets like BostonHousing).

• It will achieve this by employing a hybrid model that seamlessly combines two structural assumptions:

1. A first-order additive model (capturing main trends and low-order interactions).

2. An Active Subspace (AS) learned linear embedding (capturing high-order, non-additive interactions).

This combination is implemented via a multi-fidelity approach where a coarse additive model informs the training of a finer, dimensionally reduced GP operating within the learned active subspace.

)2. Optimized Bayesian Optimization (BO) for High Dimensions

The system can be used to accelerate expensive black-box optimization tasks in high-dimensional settings.

• The system will replace standard BO strategies that rely on Euclidean distances with a distance metric implicitly defined by the learned Active Subspace projection, drastically improving exploration efficiency.

• It will perform sequential decision-making (BO loops) much faster than baseline GPs by leveraging the structure learned from previous evaluations (via the multi-fidelity recursive formulation).

)3. Robust Uncertainty Quantification (UQ)

The system will provide significantly more reliable uncertainty estimates compared to standard GPs in high dimensions.

• When predicting unknown points, the model will output a full predictive distribution that accounts for both additive trend uncertainty and the variance associated with the learned low intrinsic dimensionality.

• It can detect when its underlying assumptions (additivity or low intrinsic dimensionality) are violated by analyzing the discrepancy between coarse and fine fidelity predictions, flagging regions where standard GPs might be over-exploring due to distance metrics.

)4. Adaptive Model Selection and Complexity Management

The system will dynamically manage model complexity based on the observed data characteristics.

• It can automatically switch between using only an additive model (when structure is strongly suspected) or a full AS-based GP (when complex interactions are required).

• It will utilize multi-fidelity sampling strategies to efficiently allocate computational budget: it will use cheap, coarse evaluations for initial trend estimation and reserve expensive, high-fidelity evaluations only for refining the specific regions where the fine model predicts high variance or significant error.

)5. Efficient Inference via Dimensionality Learning

The system can learn the essential structure of a function directly from data without requiring prior knowledge of its underlying physical constraints.

• It will incorporate a learning loop (as detailed in Algorithm 1) to estimate the intrinsic dimension and the optimal active subspace rotation matrix in real-time during training, rather than relying on fixed or random projections.