Inducing PointEdit
Inducing points are a foundational idea in scalable probabilistic modeling, particularly in the realm of Gaussian processes. By selecting a small set of representative input locations and summarizing the function values there, practitioners can approximate a large, potentially intractable problem with far simpler computations. The resulting methods enable regression and classification on datasets that would be impractical to handle with a full Gaussian process, while aiming to preserve the essential structure and uncertainty quantification that these models provide.
In practice, inducing points are not observed data. They are latent devices that capture the global behavior of a function over the input space. The induced summary can be interpreted as a low-dimensional surrogate for the latent function, and it can be optimized jointly with model parameters. Because there are several distinct families of approaches that use inducing points, the exact role of the inducing points can differ: some methods impose conditional independence given the inducing variables, while others adopt variational formulations to approximate the true posterior over the latent function.
Concept and formalism
A typical setting involves a Gaussian process over a function f with input space X, written as f ~ GP(0, k(·, ·)), where k is a kernel (covariance) function. Rather than working with the full set of function values f(X) directly, one introduces a set of m inducing inputs Z = {z1, ..., zm} in X and the corresponding function values u = f(Z). These inducing points act as a compact summary of the function’s behavior.
Two closely related notions appear in the literature. Inducing inputs refer to the locations Z in the input space, while inducing variables refer to the latent function values u = f(Z) at those locations. Many approaches build a conditional model p(f | u) that expresses the full function values given the inducing values, and then either (a) approximate the marginal likelihood p(f) by integrating over u with an independence assumption (as in FITC and PITC), or (b) optimize a variational distribution q(u) to bound the true posterior p(u | f) (as in variational inducing point methods and SVGP).
Key ideas include: - A smaller set of inducing points reduces the dimensionality of the latent space the model must manage. - The computational cost of training and prediction scales with the number of inducing points m and the number of data points n, often in the form O(n m^2) for training and O(m) or O(n m) for prediction, depending on the exact formulation. - The quality of the approximation depends on choosing informative inducing points and on how the inducing variables are integrated out or optimized.
Links to core concepts: Gaussian process, Inducing inputs, Inducing variables.
Methods that use inducing points
Fully Independent Training Conditional (FITC) and its variants represent one of the earliest practical families of inducing-point methods. They relax the exact GP likelihood by assuming conditional independence of training points given the inducing values, which yields substantial computational benefits. See Fully Independent Training Conditional and PITC for related partially independent formulations.
Variational inducing point methods introduce a variational distribution q(u) over the inducing values and derive a lower bound on the marginal likelihood that can be optimized efficiently. This framework provides principled uncertainty estimates and often leads to scalable training when combined with stochastic optimization. See Variational inference and Stochastic Variational Gaussian Process for common instantiations.
Stochastic Variational Gaussian Processes (SVGP) blend variational inducing points with stochastic optimization to scale to very large datasets. They typically maintain a set of inducing points and alternately update the variational parameters for u, the kernel hyperparameters, and the inducing point locations as needed. See Stochastic Variational Gaussian Process and the broader literature on Variational inference.
Nyström-based approaches approximate the kernel matrix using a set of landmark points, which closely mirrors the inducing-point idea. While not always identical in derivation, Nyström methods and inducing-point frameworks share the common goal of reducing computational burden while preserving essential structure of the kernel.
Links: Fully Independent Training Conditional, PITC, Stochastic Variational Gaussian Process, Variational inference, Nyström method.
Computational considerations and practical guidance
The principal motivation is scalability. Full GP inference scales cubically with the number of data points n, which is prohibitive for large n. Inducing-point methods replace this with cost that scales more gently with n, often polynomially in m and n.
The choice of m (the number of inducing points) is a central practical decision. A larger m improves approximation quality but increases cost. In practice, m is chosen based on data size, available compute, and required accuracy, and it is common to treat m as a hyperparameter to tune.
The placement of inducing points matters. Z can be fixed on a grid, selected by clustering the inputs (e.g., k-means), or treated as parameters to be optimized jointly with model hyperparameters. Variational formulations often allow joint optimization of the inducing point locations.
Inducing-point methods can be used for both regression and classification and across various likelihoods, though the exact inference procedure differs with the likelihood. See discussions around Gaussian process regression and classification, and note how variational methods extend to non-Gaussian likelihoods via bounds and approximations.
Applications, advantages, and limitations
Applications span large-scale spatial statistics, time-series modeling, and general machine learning problems where uncertainty quantification is valuable but full GP inference is infeasible.
Advantages include substantial reductions in computational burden, the ability to scale to large datasets, and the preservation of a probabilistic treatment of uncertainty.
Limitations involve approximation error from reducing to a smaller set of inducing points, sensitivity to the number and placement of inducing points, and, in some formulations, reliance on assumptions that may not hold in all data regimes. Ongoing research continues to refine inducing-point strategies, improve robustness, and compare alternative scalable GP approaches.
Links: Gaussian process, Inducing inputs, Inducing variables.