Sparse variational GP regression

For a Gaussian likelihood the optimal variational distribution is available in closed form, so the stochastic gradient loop of the paper is unnecessary and the collapsed bound can be evaluated exactly. When the inducing inputs ARE the training inputs the trace term is zero and both the bound and the predictive mean coincide with full GP regression – that identity is the anchor, and it is the one thing a sparse approximation must satisfy.

Usage

Gpsvi(
  X,
  y,
  X_test = NULL,
  inducing = NULL,
  batch_size = NULL,
  lengthscale = 1,
  variance = 1,
  noise = 0.1,
  jitter = 1e-09
)

Arguments

X

Training inputs.

y

Training responses.

X_test

Test inputs; the training inputs by default.

inducing

Inducing inputs; the training inputs by default.

batch_size

Unused placeholder kept for the interface.

lengthscale, variance, noise

GP hyperparameters.

jitter

Ridge added to K_mm.

Value

List with estimate, mean, variance, elbo, gaussian_term, trace_term, n, method.

Details

Formula: Q = K_nm K_mm^{-1} K_mn; L = log N(y; 0, Q + s2 I) - tr(K_nn - Q) / (2 s2).

References

Hensman, Fusi and Lawrence (2013), Gaussian processes for big data, UAI 2013, arXiv:1309.6835; Titsias (2009), AISTATS 2009, PMLR 5:567-574.

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
Gpsvi(V, V)
#> $estimate
#> [1] 0.9699062
#> 
#> $mean
#> [1] 0.9699062 1.9174208 2.8895238 3.8378281 4.8330643 5.7005738 6.9004118
#> [8] 7.3662141
#> 
#> $variance
#> [1] 0.08581283 0.07832638 0.07661860 0.07635845 0.07635845 0.07661860 0.07832638
#> [8] 0.08581283
#> 
#> $elbo
#> [1] -54.09351
#> 
#> $gaussian_term
#> [1] -54.09351
#> 
#> $trace_term
#> [1] 8e-09
#> 
#> $n
#> [1] 8
#> 
#> $method
#> [1] "collapsed sparse bound of Titsias (2009) as used by Hensman, Fusi & Lawrence (2013)"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"