--- kind: concept slug: gaussian-process title: Gaussian process summary: >- A Gaussian process (GP) gives a mean and a variance for each prediction, and the variance grows away from training data. Ten cards use it for uncertainty, Bayesian optimization, terrain costs, track smoothing, human motion, policy gradients, dense image matching and additive kernels. All results are author-reported. tags: [uncertainty, calibration, out-of-distribution-detection, bayesian-optimization, safe-exploration, policy-gradient, interpolation, traversability, human-robot-interaction, dense-matching, interpretability] papers: ["1112.4394", "2006.10108", "2202.13514", "2205.14090", "2305.15404", "2411.12573", "2412.07826", "2412.13474", "2412.20350", "2604.27563"] related: [uncertainty-estimation, calibration, ensembles, bayesian-optimization, bayesian-quadrature, thompson-sampling, trust-region-method, surrogate-modeling, policy-gradient, actor-critic, random-feature-approximation, spectral-normalization, visual-object-tracking, bird-eye-view-perception, foundation-model, anomaly-detection, finite-state-machine, human-in-the-loop, impedance-control, dense-correspondence, local-feature-matching, transfer-learning, additive-kernel, interpretable-machine-learning] status: draft updated_at: 2026-11-01 --- ## Overview A Gaussian process (GP) is a prior over functions. Its prediction at a new input comes with a variance. The ten cards use GPs for different tasks, and only some of them use the variance. The uncertainty use rests on an argument about distance. The SNGP paper says the variance should grow for inputs far from the training data, and calls this distance awareness [arxiv:2006.10108 §1] [arxiv:2006.10108 §2]. It places a GP prior over the training logits with an RBF kernel [arxiv:2006.10108 §3.1]. It also says an exact GP is too costly for large classification problems [arxiv:2006.10108 §3.1], and that applying a GP to high-dimensional data needs a learned feature extractor, which can destroy the distance property [arxiv:2006.10108 §1]. The other nine cards use GPs for other purposes: - **Bayesian optimization.** EGP-TS uses a weighted mixture of GPs, one per kernel in a fixed dictionary, as its surrogate [arxiv:2205.14090 §3]. HdSafeBO fits one GP to an objective and one to a safety function, and searches a latent trust region [arxiv:2412.20350 §3]. The locomotion paper fits a GP to the cost of detector thresholds and searches it with a mean-and-spread acquisition [arxiv:2411.12573 §2.5.2]. - **Online terrain model.** SALON fits a GP from visual features, speed and roughness, and converts its prediction into a risk-aware value [arxiv:2412.07826 §3.4]. - **Trajectory smoothing.** GSI fits a GP regression to each coordinate of a tracked object after filling short gaps linearly [arxiv:2202.13514 §IV-B]. - **Human motion.** HRI-MPC fits a GP regression to earlier human reaching data. It predicts where a person switches from the fast part of a reach to the careful part, which is where control is handed over [arxiv:2412.13474 §2.2]. - **Policy gradients.** BPG treats the gradient of the expected return as a Bayesian integral, with a GP prior on its integrand and Bayesian quadrature for the posterior [arxiv:2604.27563 §3] [arxiv:2604.27563 §4]. - **Dense matching.** RoMa keeps a GP as its match encoder, following DKM. The GP is a feature encoder in that paper, not an uncertainty model [arxiv:2305.15404 §3.1] [arxiv:2305.15404 app. B]. - **Interpretable regression.** The additive kernel sums products of one-dimensional kernels over all interaction orders, with one variance per order. The fitted variances show which orders carry the signal [arxiv:1112.4394 §3.2] [arxiv:1112.4394 tab. 1]. The full treatment is on the [additive kernel](/concepts/additive-kernel) page. ## Variants ### Exact GP on learned features A GP with an RBF kernel over hidden features, with the kernel matrix computed exactly. The SNGP paper rules this out for large classification problems because the kernel is too costly [arxiv:2006.10108 §3.1]. No card here runs it at scale. ### Additive kernel over interaction orders (additive GP) An exact GP whose kernel is a sum over interaction orders. The order-n term is σ_n² times the sum, over all n-subsets of inputs, of the product of their one-dimensional base kernels [arxiv:1112.4394 §3.1] [arxiv:1112.4394 eq. 1]– [arxiv:1112.4394 eq. 3]. Newton–Girard identities evaluate all orders in O(D²) [arxiv:1112.4394 §3.3] [arxiv:1112.4394 eq. 6]. The Gram matrix costs O(N²DR) for maximum order R, and the authors note that the first few orders usually matter most, so R can be lowered [arxiv:1112.4394 §3.4]. The experiments use R = 10, or D when D < 10, chosen for speed [arxiv:1112.4394 §5.2]. Hyperparameters are fitted by maximising the marginal likelihood [arxiv:1112.4394 §5.2]. Card: [Additive Gaussian Processes](/papers/1112.4394). The [additive kernel](/concepts/additive-kernel) page covers the formulas and the Newton–Girard recursion. ### Random-feature GP output layer (SNGP) The output layer becomes a GP. Its kernel is approximated by a fixed random projection followed by a cosine, so only the output weights are learned, and the posterior is an ordinary Bayesian linear model over those features [arxiv:2006.10108 §3.1]. The paper's own posterior uses a Laplace approximation, with a per-class precision accumulated over a final pass of the training data [arxiv:2006.10108 §3.1] [arxiv:2006.10108 alg. 1]. The variant also normalizes the spectral norm of the hidden weight matrices, so that each residual block is close to distance-preserving [arxiv:2006.10108 §3.2]. Card: [SNGP](/papers/2006.10108). ### Mixture of GPs for Bayesian optimization (EGP-TS) The surrogate is a weighted mixture of GPs, one component per kernel in a dictionary. The prior weight of each component is 1/M in the experiments [arxiv:2205.14090 §3] [arxiv:2205.14090 §5]. After each observation, each component's weight is multiplied by the likelihood of the new value under that component and then renormalized [arxiv:2205.14090 §3.1] [arxiv:2205.14090 eq. 13]. Each query uses Thompson sampling. The method draws one component in proportion to its weight, draws a function from that component's posterior, and queries the maximiser of that function [arxiv:2205.14090 §3] [arxiv:2205.14090 eq. 10]. Random Fourier features make the draw cheap, with 50 frequencies per component in the experiments [arxiv:2205.14090 §5] [arxiv:2205.14090 §8]. Card: [EGP-TS](/papers/2205.14090). ### Safe Bayesian optimization with GP surrogates (HdSafeBO) Two GPs, one for the objective f and one for the safety function g, are fitted with the standard posterior mean and variance [arxiv:2412.20350 eq. 2]. The search runs in a low-dimensional latent space given by PCA or an isometric VAE, with an isometric map back to the input space [arxiv:2412.20350 §4.3]. Inside a trust region around the best safe point, a candidate is kept only if its optimistic safety bound, μ + βσ, passes a threshold [arxiv:2412.20350 §4.1] [arxiv:2412.20350 alg. 1]. The paper chooses β so that Φ(β) ≤ 1 − α, which gives a per-step probability bound on safety [arxiv:2412.20350 §4.1 prop. 4.1] [arxiv:2412.20350 eq. 3]. The next point is the maximiser of one Thompson sample of f over the safe set [arxiv:2412.20350 alg. 1] [arxiv:2412.20350 §4]. The experiments use β = 2 and a Matérn kernel whose ν is not stated [arxiv:2412.20350 app. B.3]. The trust region grows after a run of safe improvements and shrinks after failures [arxiv:2412.20350 §4.2]. Card: [HdSafeBO](/papers/2412.20350). ### GP surrogate for tuning detector thresholds (locomotion BO) The objective of the threshold search is replaced by a GP surrogate. The GP has an RBF kernel, fitted with GPy, over (threshold pair, cost) samples [arxiv:2411.12573 §2.5.2]. The cost is computed from labelled transitions: it penalises a missed first transition and a wrong Walk output after that transition, and it adds a regularisation term based on each threshold's squared distance to its limit [arxiv:2411.12573 alg. 2] [arxiv:2411.12573 eq. 10]. The acquisition is k·mean(S) ± (1 − k)·std(S), with k = 0.5, and the next threshold minimises it over the search bounds [arxiv:2411.12573 eq. 8] [arxiv:2411.12573 eq. 9]. The search runs for 30 iterations, and the paper compares it with a grid search of 100 evaluations [arxiv:2411.12573 §4.3]. Card: [Locomotion mode transitions](/papers/2411.12573). ### Online terrain model with a risk measure (SALON) A GP with an RBF kernel takes a cell's visual feature and speed, and predicts the mean and variance of terrain roughness [arxiv:2412.07826 §3.4] [arxiv:2412.07826 eq. 3]. The risk-adjusted roughness adds a multiple of the variance, set by the user's risk tolerance α_R [arxiv:2412.07826 eq. 4]. The speed limit is the same kind of value, computed from a GP conditioned on a roughness cap R_max [arxiv:2412.07826 eq. 5]. The authors prefer a GP over an MLP because it adapts faster and its predictions are stable [arxiv:2412.07826 §3.4]. The training set is a buffer of samples gathered while the robot drives. The buffer is pruned by removing samples from the most common class and speed, which raises the average pairwise distance between buffer points from 3.47 to 4.79 in the first scenario [arxiv:2412.07826 §3.3] [arxiv:2412.07826 tab. 1]. Card: [SALON](/papers/2412.07826). ### GP regression for gap filling (GSI) Before smoothing, gaps shorter than 20 frames are filled linearly [arxiv:2202.13514 §V-A]. Each track's x, y, w and h are then smoothed by a GP with an RBF kernel. The length scale depends on the track length l as λ = τ · log(τ³ / l) with τ = 10 [arxiv:2202.13514 §IV-B] [arxiv:2202.13514 eq. 15]. The smoothed values are the posterior mean at the track's frames [arxiv:2202.13514 §IV-B] [arxiv:2202.13514 eq. 14]. The paper gives no noise value for this regression [arxiv:2202.13514 §IV-B]. Card: [StrongSORT](/papers/2202.13514). ### GP regression for handover timing (HRI-MPC) A GP regression is fitted to earlier experimental data and predicts the position of the transition between the two sub-movements of a reach. Its inputs are the distance to the goal and the goal size, normalised by arm length [arxiv:2412.13474 §2.2] [arxiv:2412.13474 fig. 2]. In the handover task the robot keeps 400 N/m of translational stiffness until that point, then sets it to 0 [arxiv:2412.13474 §5.2]. The kernel and the training split are not stated. Card: [HRI-MPC](/papers/2412.13474). ### Exact GP posterior over policy gradients (BPG) The gradient is a GP posterior over the score-function integrand. With the Fisher kernel k(ξ, ξ') = uᵀ G⁻¹ u', the posterior mean is B C y and the covariance is G − B C Bᵀ, where C = (K + Σ)⁻¹ is formed over the sampled paths [arxiv:2604.27563 §4.2] [arxiv:2604.27563 prop. 4] [arxiv:2604.27563 alg. 1]. A second model uses the quadratic Fisher kernel (1 + uᵀ G⁻¹ u')², which makes the covariance a scalar times the identity, with b₀ = 1 + n for n policy parameters [arxiv:2604.27563 §4.1] [arxiv:2604.27563 prop. 3]. A natural-gradient step multiplies the posterior mean by G⁻¹ [arxiv:2604.27563 alg. 2]. Card: [BPG and BAC](/papers/2604.27563). ### GPTD critic (BAC) The critic is a GP over the state-action value function, fitted with GPTD. The GPTD model treats each reward as Q(z_i) − γ Q(z_{i+1}) plus noise [arxiv:2604.27563 §7.1]. The kernel adds a state kernel to a Fisher kernel on state-action pairs [arxiv:2604.27563 §7.2]. The gradient posterior then uses the score vectors of the visited pairs and the critic's weights [arxiv:2604.27563 §7.2] [arxiv:2604.27563 prop. 6]. Card: [BPG and BAC](/papers/2604.27563). ### GP match encoder in dense matching (RoMa) RoMa's match encoder is a GP, carried over from DKM. It works on the frozen coarse features and outputs 512-dimensional embeddings, using an exponential cosine kernel [arxiv:2305.15404 §3.1] [arxiv:2305.15404 app. B]. The appendix describes the GP output as a posterior over coordinates in the second image's embedding [arxiv:2305.15404 app. B]. A Transformer decoder reads the GP output with the DINOv2 features and predicts logits over 64 × 64 anchors plus one matchability logit, instead of regressing the warp directly [arxiv:2305.15404 §3.3] [arxiv:2305.15404 app. B]. The ablation has no row that removes the GP alone, so its own contribution is not measured [arxiv:2305.15404 tab. 2]. The paper gives an inverse temperature of 10 for the kernel; the released code uses a different parameterisation (see the [RoMa card](/papers/2305.15404) and Contradictions). Card: [RoMa](/papers/2305.15404). ## When to use which The results are author-reported. The SNGP rows come from one paper's tables. The paper does not give the hardware for each latency row [arxiv:2006.10108 tab. 2] [arxiv:2006.10108 tab. 4]. The EGP-TS results are plots, not numbers in the text [arxiv:2205.14090 §5]. The SALON, HRI-MPC and locomotion results come from one robot or one operator each. | Situation | Use | Why | |---|---|---| | You can retrain the network and need uncertainty from one model | Random-feature GP output layer (SNGP) | Accuracy and calibration are reported as close to a 10-model ensemble, at about 6 ms per example versus 38 ms for the ensemble on CIFAR-10 [arxiv:2006.10108 tab. 2] | | Corrupted inputs matter most and you can afford several models | Ensemble | On corrupted CIFAR-10 the 10-model ensemble has the best accuracy, 77.9 against 74.6 for SNGP [arxiv:2006.10108 tab. 2] | | Regression or classification on a few tabular inputs, where single inputs and small groups of inputs are plausible drivers and the per-order variances should be read | Additive kernel (additive GP) | Regression MSE on concrete is 0.089, against 0.157 for an SE-GP with ARD and 0.147 for HKL [arxiv:1112.4394 tab. 2]. On housing the SE-GP is slightly better, 0.092 against 0.102 [arxiv:1112.4394 tab. 2]. Classification is mixed: the SE-GP has lower error on breast and ionosphere, the additive GP on sonar [arxiv:1112.4394 tab. 4]. The authors report that no competitor significantly beats it [arxiv:1112.4394 §5.3]. The results cover five regression and six classification datasets | | You cannot afford an extra pass over the training set at the end of training | Not SNGP as written | The precision update runs that pass at the last epoch [arxiv:2006.10108 alg. 1] | | You need an exact GP posterior on small or low-dimensional inputs | Not covered by a card | The paper's cost argument concerns large classification problems [arxiv:2006.10108 §3.1] | | You cannot modify hidden layers | Not covered | The method changes training (spectral normalization) and the output layer [arxiv:2006.10108 §3.2] | | Black-box tuning where the kernel is not known in advance and evaluations are expensive | Mixture of GPs with Thompson sampling (EGP-TS) | The authors report lower simple regret than single-kernel GP Thompson sampling, GP-EI and two bandit ensembles on synthetic functions and robot pushing [arxiv:2205.14090 §5.1] [arxiv:2205.14090 §5.2]. The regret bound covers the exact-kernel method, not the random-feature version used in practice [arxiv:2205.14090 §4] | | A controller with dozens to thousands of parameters, and safety must hold during every trial | Safe BO with GP surrogates (HdSafeBO) | On 1000-D synthetic GP functions, the objective is 3.96 against 3.07 for LineBO, and cumulative violation is 27.42 against 36.59 [arxiv:2412.20350 tab. 1]. The paper says SafeOpt may fail to grow its safe region once there are about 10 parameters [arxiv:2412.20350 §1]. Safety depends on the embedding and on the unstated probability level α, and the authors warn that optimism can cause unsafe choices in unexplored problems [arxiv:2412.20350 §6] | | A few thresholds per user, labelled trials from that user, and no real-time requirement | GP-surrogate BO of thresholds (locomotion) | Supervised BO raises the autonomyo accuracies to 90 / 90 / 87.5 / 82.5 % from 77.5 / 75 / 77.5 / 75 % [arxiv:2411.12573 §4.2] [arxiv:2411.12573 fig. 6]. It needs ground-truth transition labels [arxiv:2411.12573 alg. 2], was not run in real time [arxiv:2411.12573 §5.5], and took about 2 minutes on a PC [arxiv:2411.12573 §5.4] | | Unseen terrain must be costed after seconds of the robot's own driving, and an operator can click one object | GP with CVaR risk (SALON) | On one ATV figure-8 course, SALON at R_max = 0.2 had 3 interventions and 2 undesirable behaviours. Velociraptor had 4 and 8, and the geometric cost had 23 and 7 [arxiv:2412.07826 tab. 4]. The numbers are one course, one driver and no variance [arxiv:2412.07826 §4.2.3] | | A handover point must be predicted from reach distance and goal size | GP regression on earlier data (HRI-MPC) | Results are qualitative: peak dispersion near the predicted point, with no numbers in the text [arxiv:2412.13474 §5.2] [arxiv:2412.13474 fig. 9]. There is one operator and five trials per goal [arxiv:2412.13474 §5.1] | | Dense two-view matching under strong appearance change, with dense supervised correspondences available for training | GP match encoder with a Transformer decoder (RoMa) | Author-reported: WxBS mAA@10px 80.1 against 58.9 for DKM [arxiv:2305.15404 tab. 4], and IMC2022 mAA@10 88.0 against 83.1 for DKM [arxiv:2305.15404 tab. 3]. At 560×560 it takes about 7% longer than DKM, 198.8 ms against 186.3 ms per pair [arxiv:2305.15404 §4.5]. Training needs dense correspondences from MVS depth or RGB-D, which the authors name as a limitation [arxiv:2305.15404 §4.2] | | Gaps of up to 19 frames in tracked boxes, with smooth motion | GP smoothing (GSI) | The table reports only the combined gain: with AFLink, StrongSORTv6 gains 1.0 IDF1 and 1.2 HOTA on MOT17 validation [arxiv:2202.13514 tab. II]. The smoothing also changes observed boxes, not only the gaps [arxiv:2202.13514 §IV-B] | | Policy gradient with few sampled episodes per update, and fixed-length episodes | BPG | On a bandit, the Bayesian estimate has about an order of magnitude less variance than Monte Carlo at M = 10 [arxiv:2604.27563 §6.1] [arxiv:2604.27563 tab. 2]. The LQR results also favour BPG over Monte Carlo for M = 10 to 40 [arxiv:2604.27563 §6.2.2] | | Trajectory lengths vary, or updates must use single transitions | BAC (GPTD critic) | The authors say BQ degrades when trajectories differ in length, and they use BAC for mountain car and ship steering [arxiv:2604.27563 §8]. BAC needs a state kernel and a Fisher estimate [arxiv:2604.27563 §7.2] | | The Fisher matrix must be estimated from data | Either, with caution | With an estimated G the closed-form identities are approximations, and the covariance is no longer exact [arxiv:2604.27563 app. A] [arxiv:2604.27563 §4.3] | ## Implementation sketch The common core is the GP posterior mean and variance. The mean is a kernel-weighted combination of the observed values, and the variance shrinks near the data [arxiv:2202.13514 eq. 14]. The variance line is the standard GP predictive form; the cards name the quantities, but none writes this line out. ```pseudocode # GP regression posterior at test inputs [arxiv:2202.13514 eq. 14] # Kernel k(x, x'); noise variance sigma2 (not stated in the GSI paper) def gp_posterior(x_train, y_train, x_test, sigma2): # 1. Gram matrices of training and test inputs K = kernel_matrix(x_train, x_train) # (N, N) k_star = kernel_matrix(x_test, x_train) # (T, N) # 2. Invert the noisy Gram matrix once inv = inverse(K + sigma2 * identity(N)) # (N, N) # 3. Posterior mean [eq. 14] mean = k_star @ inv @ y_train # (T,) # 4. Posterior variance, prior minus what the data explain var = kernel_diag(x_test) - rowsum((k_star @ inv) * k_star) # (T,) return mean, var ``` The SNGP variant replaces the exact kernel with random features, so the prediction is one forward pass plus a quadratic form. Where the paper's equations disagree, the sketch uses the form from the appendix; see Contradictions. ```pseudocode # Random-feature GP output layer, prediction for one input # Frozen random projection W_L (D_L, D_h) and bias b_L (D_L,) [arxiv:2006.10108 §3.1] # Learned: output weights beta (D_L, K) and per-class covariance [alg. 1, alg. 2] def gp_predict(x): # 1. Hidden features from the network body [§3.2] h = hidden_network(x) # (D_h,) # 2. Random Fourier features approximating an RBF kernel [§3.1] phi = sqrt(2 / D_L) * cos(W_L @ h + b_L) # (D_L,) # 3. Posterior mean of each class logit [alg. 2] mean = phi @ beta # (K,) # 4. Posterior variance of each class logit [alg. 2] var = [phi @ cov[k] @ phi for k in range(K)] # (K,) return mean, var ``` ## Contradictions Across the ten cards, no two papers report conflicting numbers for the same quantity. They measure different things (OOD AUPR and calibration, simple regret, safe objective and violation, tracking HOTA and IDF1, policy learning curves, exoskeleton transition accuracy, navigation interventions, handover timing, dense-matching mAA and AUC), so their numbers should not be compared directly. Within and around individual papers: - **Additive GP optimizer and initialisation.** The paper gives L-BFGS for 500 iterations [arxiv:1112.4394 §5.2]. The card reads the released code as passing a function-evaluation budget instead, and as storing the initial order variance so that it is squared on use. These readings come from the code, not from runs. See the card's Paper vs code section before reusing the paper's settings [Additive Gaussian Processes](/papers/1112.4394). - **SNGP precision update.** §3.1 and appendix C give the minibatch update with the weights on the previous and new precision swapped. With the paper's discount factor of 0.999 the two forms differ strongly [arxiv:2006.10108 §3.1] [arxiv:2006.10108 app. C]. - **SNGP initial precision.** Algorithm 1 starts the precision at the identity; appendix C starts it at a scaled identity with a small factor [arxiv:2006.10108 alg. 1] [arxiv:2006.10108 app. C]. - **SNGP sign of the projection.** §3.1 writes the projection with a minus sign on the weights, while algorithm 2 does not [arxiv:2006.10108 §3.1] [arxiv:2006.10108 alg. 2]. The weights are frozen, so either sign gives a valid model if it is used consistently in training and prediction; mixing them does not. - **SNGP prediction method.** The paper's results average 10 Monte Carlo samples of the logits [arxiv:2006.10108 §5]. The released code's default is a mean-field approximation with a tunable factor that the paper does not state [arxiv:2006.10108 app. A]. The card records the code-side details in full at [the card](/papers/2006.10108). - **SNGP learning rate for CIFAR-100.** Appendix C gives 0.08 [arxiv:2006.10108 app. C]. The released code's default is also 0.08, but its docstring tells CIFAR-100 users to set 0.04 (see the card's Paper vs code section). Check the run configuration before reusing either value. - **SNGP OOD score.** The CIFAR tables use the Dempster-Shafer uncertainty of the logits [arxiv:2006.10108 app. C]. The released code marks the Monte Carlo version of that score as degenerate and falls back to a different score by default [arxiv:2006.10108 app. C]. Compare OOD numbers only with the same score. - **EGP-TS refits.** The paper refits hyperparameters and weights every 50 iterations [arxiv:2205.14090 §5]. The released notebook turns refits off by default, and its random-feature scaling does not match the RBF length scale (see the card's Paper vs code section). Results from the notebook as released may not reproduce the paper's settings. - **EGP-TS regret theory.** The bound is sublinear for the exact-kernel method [arxiv:2205.14090 §4]. Whether it holds for the random-feature sampler used in the experiments is not covered by the paper. - **HdSafeBO safety threshold.** Section 3 defines a safe point as g > 0. The candidate filter in algorithm 1 compares μ + βσ with a threshold h, while the update in algorithm 2 uses 0 [arxiv:2412.20350 alg. 1] [arxiv:2412.20350 alg. 2]. The experiments use different thresholds in different units [arxiv:2412.20350 app. B.4] [arxiv:2412.20350 app. B.5]. Choose one convention and apply it everywhere. - **HdSafeBO bound scale in code.** The paper's bound is μ + βσ [arxiv:2412.20350 §3]. The released code computes the bound through a library call whose scale the card could not check, so the effective multiplier may differ. See the card's Paper vs code section before reusing either form. - **HdSafeBO trust region.** The paper does not give the trust-region constants [arxiv:2412.20350 app. B.3]. The released code's restart value and success rule differ from the algorithm (see the card's Paper vs code section), so the trust region may shrink in practice where the paper expects it to grow. - **HdSafeBO guarantee.** The per-step probability bound depends on α, which the paper does not state [arxiv:2412.20350 §4.1]. The guarantee also assumes the embedding is accurate; the IRVAE's linear distance correlation is 0.9729, so the isometry does not strictly hold [arxiv:2412.20350 tab. 4] [arxiv:2412.20350 app. A.3]. - **Locomotion acquisition sign.** The acquisition is written with ± on the standard deviation, and the paper does not state which sign it uses [arxiv:2411.12573 eq. 8] [arxiv:2411.12573 §2.5.2]. - **Locomotion subject S2.** Results say S2's W-S was missed before BO and detected after [arxiv:2411.12573 §4.2]. The discussion says S2's W-S and S-W improved from 80% and 60% to 100% [arxiv:2411.12573 §5.3]. The two statements do not agree. - **Locomotion penalty shape.** The text applies the limit penalty when a threshold exceeds 55° or falls below 5°, while the equation penalises the squared distance to the limit for every component [arxiv:2411.12573 §2.5.2] [arxiv:2411.12573 eq. 10]. - **SALON online fitting.** The paper describes an RBF GP without online refitting [arxiv:2412.07826 §3.4]. The released code refits the GP every 10 cycles and fine-tunes the speed GP with SGD (see the card's Paper vs code section). The speed GP also uses a subset of the feature columns, while the paper conditions speed on the feature and on R_max [arxiv:2412.07826 eq. 5]. - **GSI length scale.** Equation 15 has no bounds [arxiv:2202.13514 eq. 15]. The released code clips the length scale to [0.1, 100] (see the card's Paper vs code section), so the implemented kernel differs from the formula for very long or very short tracks. - **BPG covariance with estimated Fisher.** Proposition 3 holds exactly only for the true Fisher matrix [arxiv:2604.27563 app. A]. The paper estimates it in its experiments [arxiv:2604.27563 §4.3] [arxiv:2604.27563 §7.2], so the reported covariances are approximate. - **RoMa kernel temperature.** The paper gives an inverse temperature of 10 for the exponential cosine kernel [arxiv:2305.15404 app. B]. The released code sets T = 0.2 and computes exp((cos − 1)/T), which the card reads as an inverse temperature of 5 (see the [RoMa card](/papers/2305.15404)). Check which convention a reimplementation uses. Not stated in the BPG paper: the noise matrix Σ for noise-free rewards. For the second model the Gram matrix has rank at most n, so it is singular for M > n unless a nugget is added. This follows from the kernel form, and the paper gives no value [arxiv:2604.27563 §4.2] [arxiv:2604.27563 §6.2.1]. ## Papers - [Additive Gaussian Processes](/papers/1112.4394) (arXiv:1112.4394): a GP whose kernel sums products of one-dimensional kernels over all interaction orders, with one variance per order, compared with linear regression, GP-GAM, HKL and an SE-GP with ARD on regression and classification benchmarks. - [SNGP](/papers/2006.10108) (arXiv:2006.10108): a deterministic network with spectral-normalized hidden layers and a random-feature GP output layer, compared with deep ensembles on CIFAR and CLINC OOS. - [EGP-TS](/papers/2205.14090) (arXiv:2205.14090): Bayesian optimization with a weighted mixture of GPs and Thompson sampling via random features, compared with single-GP and bandit baselines on synthetic functions and robot pushing. - [HdSafeBO](/papers/2412.20350) (arXiv:2412.20350): safe Bayesian optimization of controllers with up to thousands of parameters, using GP surrogates for objective and safety in a latent trust region, compared with LineBO, SCBO, CONFIG, cEI and CMA-ES on synthetic functions. - [Locomotion mode transitions](/papers/2411.12573) (arXiv:2411.12573): personalizing a threshold-based transition detector on two exoskeletons, with GP-surrogate Bayesian optimization of thresholds against labelled user data, compared with grid search and a statistics-based rescaling. - [SALON](/papers/2412.07826) (arXiv:2412.07826): online off-road traversability learning with a GP over foundation-model features and bandpower roughness, with CVaR-based cost and speed maps, compared with geometric, GANav, HDIF and Velociraptor costs. - [StrongSORT](/papers/2202.13514) (arXiv:2202.13514): a tracker whose optional GSI step smooths each track with GP regression to fill gaps, reported with AFLink. - [HRI-MPC](/papers/2412.13474) (arXiv:2412.13474): planning for human-robot co-manipulation with a human motor-control objective, using a GP regression to predict the handover point, with qualitative results on a Franka arm. - [Bayesian policy gradient and actor-critic](/papers/2604.27563) (arXiv:2604.27563): GP and Bayesian-quadrature estimates of the policy gradient (BPG), and a GPTD critic (BAC), compared with Monte Carlo policy gradient. - [RoMa](/papers/2305.15404) (arXiv:2305.15404): dense two-view feature matching with a GP match encoder, a Transformer anchor decoder and a robust refinement loss, compared with DKM, LoFTR, ASpanFormer and others on MegaDepth, ScanNet, WxBS, IMC2022 and InLoc.