Kernel Learning for Sample Constrained Black-Box Optimization

Rajalaxmi Rajagopalan; Romit Roy Choudhury; Yu-Lin Wei

REVIEW 5 major objections 5 minor 15 references

Kernel Learning for Sample Constrained Black-Box Optimization

T0 review · 5 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Kernel selection in black-box optimization can be recast as a black-box optimization problem in the continuous latent space of a VAE, and doing so reaches optima in far fewer function evaluations.

desk verdict A genuinely novel VAE-based kernel selection mechanism with plausible synthetic gains, but the continuity/completeness assumption that carries the method is asserted rather than tested, and the real-world evidence is too thin to confirm the headline claim. read the letter →

arxiv 2507.20533 v1 pith:IZ7OXM3U submitted 2025-07-28 cs.LG

Rajalaxmi Rajagopalan , Yu-Lin Wei , Romit Roy Choudhury This is my paper · ORCID

classification cs.LG

keywords black-boxoptimizationkernellearningGaussianprocessregressionvariationalautoencoderBayesiancompositekernelssampleefficiencyaudiopersonalization

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Black-box optimization assumes each evaluation of the unknown function is costly, so the paper asks whether the surrogate model's kernel can be learned from the same samples the optimizer already collects. It proposes KOBO, which turns the discrete space of composite kernels into a continuous latent space of a variational autoencoder, then runs a second Gaussian-process search in that latent space to maximize model evidence: how plausible the observed samples are under a candidate kernel. The paper claims this finds the right kernel shape earlier than greedy or random kernel searches, so the main optimizer's surrogate matches the function's structure and the optimum is reached with fewer queries. On a discontinuous staircase benchmark, KOBO reaches the global minimum in about 17 function evaluations, versus 28 for MCMC search, 32 for BOMS, and 43 for CKS. The same pipeline is demonstrated on hearing-aid filter personalization with real users and on rating-based image recommendation.

What carries the argument

The load-bearing object is the continuous latent kernel space $\mathcal{Z}$ learned by KerVAE. KerVAE is trained on composite kernels encoded as $r = [r_c, r_d]$: $r_c$ is a 15-slot grammar vector recording which base kernels (squared-exponential, periodic, rational-quadratic, Matérn, linear) are added or multiplied, and $r_d$ is the vector of Frobenius distances between the composite kernel's covariance matrix and each base kernel's covariance matrix on the current observations. The continuity of this representation is what lets KerGPR treat model evidence as a smooth black-box function and optimize it with a standard GP, instead of exhaustively expanding a discrete kernel tree. The decoder maps the resulting optimal latent point back to an interpretable composite kernel for the function-space GPR.

What would settle it

Take a target function drawn from a known composite kernel that is in the training grammar, run KOBO to a fixed budget, and compare the decoded $K^*$ with the true kernel and with the best kernel found by exhaustive enumeration in that grammar. If the decoded kernel either fails to match the true one or is outperformed by a grammar kernel that KerGPR never approaches, the latent map is not faithful and the sample-efficiency claim would not generalize.

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Extended reading notes

Core claim

The paper's central claim is that kernel selection for Gaussian-process black-box optimization can be treated as a black-box optimization problem in a continuous latent space rather than as a search over a discrete grammar. Composite kernels are described by a grammar-based code $r_c$ concatenated with a data-based code $r_d$ that records Frobenius distances between the composite kernel's covariance matrix and those of the base kernels on observed samples. A variational autoencoder (KerVAE) maps this coded kernel space to a low-dimensional continuous latent space $\mathcal{Z}$; a second Gaussian process (KerGPR) maximizes model evidence $L(K) = P(F|X,K)$ over $\mathcal{Z}$, and the decoder turns the best latent point into the kernel $K^*$ that the main function-space GPR uses. The paper reports that $K^*$ converges to ground-truth kernels when the target function is drawn from a known GP kernel, that it recovers periodic-plus-trend structure in real CO2 data, and that it lowers regret faster than the MCMC, BOMS, and CKS baselines.

Load-bearing premise

The argument collapses if the VAE latent space is not continuous and complete: nearby latent points must decode to similar kernels and every latent point must decode to a valid kernel, because KerGPR's optimum is only as good as the map from Z back to kernels.

Editorial extensions

If this is right

For optimization problems whose hidden function has flat plateaus or periodic trends, learned composite kernels should reach better or equal solutions within a fixed query budget than any single base kernel, because the surrogate can represent the same structure.
The auxiliary kernel search runs in latent space and does not consume user or function-evaluation budget, so the sample-efficiency gain should persist even when KerVAE retraining is frequent.
Decoded $K^*$ remains interpretable as a grammar composition (e.g., $SE \times PER + RQ$), so the method doubles as structure discovery: it names the function's components, not just a fitted matrix.
If a target function's true kernel is expressible in the grammar, KOBO should recover it exactly as sample count grows, as the paper's ground-truth kernel experiments indicate.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the latent space generalizes, the same encode-decode-and-optimize loop is a template for other discrete model-selection problems, such as choosing acquisition functions or prompting strategies, wherever a model-evidence-like score can be computed.
A practical extension would be to freeze KerVAE after training and only recompute the data-based part $r_d$ for each new objective, making kernel selection nearly free at deployment time.
One testable risk: if the latent space contains regions that decode to degenerate or invalid kernels, KerGPR might waste latent evaluations there; a dense-coverage sanity check that samples many latent points and measures reconstruction validity and evidence smoothness would reveal whether such dead regions limit the reported gains.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Worth a look if you care about kernel learning for BO. The core idea is fresh: instead of searching a discrete grammar space of composite kernels, KOBO trains a VAE on grammar-based kernel codes plus a data-dependent distance feature, then runs a second GP in the 2D latent space to pick kernels for the main optimizer. I haven't seen that exact combination before.

What I liked: the ground-truth kernel recovery check in Table 1 is a good sanity test, and the CO2 example shows the learned kernels track structure (periodic, trend) in a way that static kernels miss. The limitations section is honest—overfitting, computational overhead, latent interpretability are all acknowledged. The synthetic comparisons against MCMC, BOMS, and CKS are plausible, and the paper is clearly written.

Soft spots, in proportion. First, every headline number is an average of 10 runs with no error bars or significance tests. The central comparison—17 vs 28 evaluations on Staircase—could be noise, and with no code or data released, I can't check. That's the biggest concrete issue. Second, the load-bearing assumption in Section 3—that the VAE latent space is continuous and complete—is never actually tested. The reconstruction check in Appendix Figure 9 only measures error on training codes, not on arbitrary latent points. Nothing samples a random z, decodes it, and verifies the result is PSD or that model evidence is smooth in z. KerGPR uses an SE kernel on that latent space, so if there are dead regions or discontinuities, the method silently fails. The stress-test note is right about this gap. It may be fixable with an experiment, but as written it's an unsupported axiom. Third, the real-world studies are thin: only three users in the main text, no SOTA kernel-search baselines there, and the image results are qualitative. Finally, the baseline setup is underspecified—do MCMC/CKS/BOMS also run on the ALEBO embedding? If not, the comparison may be apples to oranges.

None of this is fatal. The idea deserves a serious referee and the authors can address the concerns. But it's a conditional accept at best in current form.

Recommendation: send to peer review. I'd want to see error bars, a latent-space validity/smoothness ablation, and a clearer baseline setup before believing the sample-efficiency claim.

Referee Report

5 major / 5 minor

Summary. The paper proposes KOBO, a Gaussian-process kernel learning method for black-box optimization under tight sample budgets. KOBO defines a discrete grammar-based space of composite kernels, augments each kernel's grammar code with a data-dependent distance vector, trains a variational autoencoder (KerVAE) to map this space into a low-dimensional continuous latent space, and runs a second GP (KerGPR) over the latent space to maximize model evidence. The resulting kernel is used by the main function-level GP. Experiments compare KOBO with static base kernels and with MCMC, CKS, and BOMS on synthetic benchmarks, a CO2 time-series structure-learning task, a hearing-aid personalization study, and a prompt-based image recommendation study.

Significance. If the method works as claimed, KOBO is a useful contribution to sample-constrained BO: it replaces discrete kernel search with continuous optimization, includes a sanity check for ground-truth kernel recovery (Table 1), an ablation of kernel encoding length, and an honest limitations section, and it reports a public audio demo URL. However, the empirical support is incomplete in several load-bearing places, and the central latent-space continuity/completeness assumption is asserted rather than demonstrated. The idea of embedding a grammar-generated kernel space into a continuous VAE latent space is reasonable and potentially valuable, but the paper currently does not provide sufficient evidence that the proposed mechanism, rather than the specific experimental setup, is responsible for the reported gains.

major comments (5)

[Section 3, 'Intuition and Overview', and Eq. (8)] The central mechanism assumes that the KerVAE latent space is continuous and complete: 'two close points in the latent space cannot decode to completely different results, and a point sampled from the latent space must decode to a valid result.' This assumption is load-bearing because KerGPR with an SE kernel optimizes model evidence over Z, implicitly assuming L(z) is smooth and well-defined everywhere. The paper provides no experiment that samples arbitrary latent points and checks PSD-validity of the decoded kernels or smoothness of the model-evidence surface. Appendix Figure 9 only measures reconstruction error on training kernel codes; it does not test untrained or dead regions of the latent space. Without such a validation, the claimed sample-efficiency gains (e.g., 17 vs. 28 function evaluations on Staircase in Section 4) cannot be confidently attributed to the continuous-space mechanism. Please add an explicit latent-space validity/smoothness study, or state conditions under which the VAE guarantees these properties.
[Table 1 and Section 4, 'Is K* indeed learning the structure of f(x)?'] The ground-truth kernel recovery test is presented as evidence that KOBO learns the structure of the objective function, but the table shows only partial recovery. For f1, the learned kernel at Q=25 is A*A*B + C*D instead of the ground-truth A*A*B + C; for f2, the learned kernel is A*C + D instead of C + D. Only f3 is recovered exactly (up to commutativity of multiplication). The limitations paragraph mentions overfitting for f1, but not the persistent mismatch for f2. Since this table is the primary sanity check for the core claim, the text should either report the recovery rate across runs, acknowledge the imperfect recovery more prominently, or demonstrate that the mismatched kernels still yield equivalent model evidence on the observed data.
[Section 4 and Technical Appendix, 'User Experiment: Audio Personalization'] The abstract claims 'real-world audio experiments with U=6 users,' but Table 2 reports results for only U1, U2, and U3, and the appendix, despite stating that six volunteers were recruited, does not present the remaining three users' data. The main text says 'rest in Appendix,' but no such table appears in the submitted material. Similarly, the image recommendation study (Table 3 and Table 5) is qualitative, with no quantitative metric or inter-rater protocol. Thus the real-world claims that KOBO personalizes hearing aids with fewer queries or converges to desirable images with limited ratings are not fully supported by the reported data. Please either include the complete user data or revise the claims to match what is actually shown.
[Section 4 and Technical Appendix, 'Synthetic Function Evaluation Parameters'] All numerical comparisons are reported as averages over 10 runs, but no error bars, standard deviations, confidence intervals, or significance tests are provided anywhere in the paper. For example, the Staircase comparison reports 17 function evaluations for KOBO versus 28 for MCMC; without variance information it is impossible to assess whether this difference is robust across random initializations and ALEBO projections. Given that the paper's central claim is a quantitative reduction in sample budget, the lack of uncertainty quantification is a substantive gap. Please add error bars or statistical tests to all main results.
[Eq. (4) and Section 4, 'Another Metric'] The kernel-selection objective in Eq. (8) is defined as exact model evidence P(F|X, Dec(z)), but the evaluation replaces it with a BIC approximation, log P(F|X,K) = -1/2 f^T K^{-1} f - 1/2 log((2*pi)^N |K|). In the low-sample regime used throughout the paper (B <= 25), the BIC approximation may be inaccurate, and KerGPR may therefore be optimizing a different objective than the one stated. The relationship between BIC-optimal and exact-marginal-likelihood-optimal kernels in this regime is not discussed. Please justify the BIC approximation for the sample sizes used, or report whether exact marginal likelihood gives similar results.

minor comments (5)

[Section 3, Kernel Combiner] The grammar code in Eq. (5) allows fractional exponents, and the appendix restricts the sum of exponents to 0 <= sum <= 3, but it does not state whether individual exponents can be negative. Negative powers of base kernels are generally not positive semi-definite, so the validity of decoded kernels is unclear; please clarify the allowed range of individual exponents.
[Section 2, Eq. (1)] The objective in Eq. (1) is written as argmin ||f(hat x) - f(x*)||_2, but f(x*) is unknown. The paper later uses regret f(hat x) - f(x*) as the evaluation metric. Please align the problem formulation with the metric actually used.
[Section 4 and Appendix] There are several typos and reference inconsistencies: 'MICHALEWICZ' should be 'Michalewicz'; 'Helliger distance' should be 'Hellinger distance'; 'Forbenius norm' should be 'Frobenius norm'; and 'Figure 5(a) and (b) in the Appendix' is incorrect because Figure 5 appears in the main text and the appendix figures are numbered 7-10.
[Section 4, Table 3] The image recommendation results in Table 3 do not specify how many raters were used, how scores were collected, or how the qualitative match to the prompt was assessed. Please describe the rating protocol and the number of users.
[Technical Appendix, 'KerVAE Reconstruction'] Figure 9 shows reconstruction error only on the training kernel codes; please also report reconstruction error on held-out kernel codes, which would be a more meaningful check of generalization.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: KOBO's kernel selection is standard empirical model selection, and its performance claims are evaluated against independent true-function regret and ground-truth kernels.

full rationale

The paper's derivation chain is self-contained and does not reduce to its own inputs. KOBO's kernel selection objective (Eq. 8) defines K* as the kernel that maximizes model evidence P(F|X, Dec(z)) on the observed data; this is standard empirical Bayesian model selection, not a disguised prediction of the same quantity. The central performance claims are measured by Regret against the true function minimum, which is external to the kernel-selection objective and cannot be forced by construction. The ground-truth kernel recovery experiment generates synthetic functions from a known GP kernel K+ and checks whether KOBO's selected kernel approaches K+; this is an external identifiability check, not a definitional equivalence, because K+ is not an input to the optimizer. The comparisons to MCMC, CKS, and BOMS are legitimate: all methods search the same kernel space using the same model-evidence objective, so comparing achieved model evidence or true-function regret is a meaningful empirical benchmark. The paper contains no load-bearing self-citations and no fitted parameter that is later renamed as a prediction. The latent-space continuity and completeness assumptions are correctness risks rather than circularity: they are asserted design expectations, not conclusions derived from the method's own outputs.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

No new physical or conceptual entities are introduced. The latent kernel space is a learned representation of existing kernel compositions, not a newly postulated object. The main load-bearing assumptions are the continuity and completeness of the VAE latent space, the sparsity/embedding assumption, and the adequacy of the BIC-approximated model evidence as a kernel selection objective.

free parameters (5)

Grammar encoding length L = 15
Fixed at L=15 after an ablation comparing L=5, 15, and 25 on the staircase function; larger L gave diminishing returns and L=5 was too weak. This caps the number of terms KOBO can represent.
KerVAE latent dimension = 2
The continuous kernel space is set to 2 dimensions with no sensitivity analysis reported; a hand-picked capacity for the latent manifold.
Maximum exponent sum per term = 3
Restricts kernel complexity in the grammar code (0 <= ai+bi+ci+di+ei <= 3); chosen ad hoc, no ablation shown.
KerGPR iterations per kernel update = 20
The kernel-space GPR runs for 20 latent evaluations before returning the current best kernel; no sensitivity analysis is provided.
ALEBO embedding dimension = 20
All high-dimensional benchmarks (N=2000) are projected to a 20-dimensional space via random embeddings; this constrains the searchable subspace and follows ALEBO's setup.

assumptions (4)

ad hoc to paper VAE latent space is continuous and complete
Section 3 assumes nearby latent points decode to similar kernels and every latent point decodes to a valid kernel. This is an expected property of VAEs, not a proven guarantee for the grammar-code space, and it is load-bearing for KerGPR's latent-space optimization.
domain assumption Random embedding preserves the minimizer (low effective dimensionality)
The Technical Appendix states the objective function is assumed sparse and ALEBO's random linear embedding must map the true minimizer to a point in the low-dimensional space. If this fails, the high-dimensional benchmarks lose meaning for all methods.
standard math Grammar compositions of base PSD kernels are PSD
Section 3 relies on closure of positive semi-definite kernels under addition and multiplication to restrict the search to the grammar-generated space K subset S.
domain assumption Model evidence with BIC approximation is a valid selection objective
Equation 4 defines L(K)=P(F|X,K) and uses a BIC approximation because exact marginal likelihood is intractable; the approximation is assumed adequate for comparing kernels.

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Cite this review

Pith. "Pith review of Kernel Learning for Sample Constrained Black-Box Optimization." pith.science (2026). https://pith.science/paper/IZ7OXM3U

@misc{pith2026250720533,
  author       = {Pith},
  title        = {Pith review of: Kernel Learning for Sample Constrained Black-Box Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IZ7OXM3U}},
  note         = {Machine review of arXiv:2507.20533}
}

read the original abstract

Black box optimization (BBO) focuses on optimizing unknown functions in high-dimensional spaces. In many applications, sampling the unknown function is expensive, imposing a tight sample budget. Ongoing work is making progress on reducing the sample budget by learning the shape/structure of the function, known as kernel learning. We propose a new method to learn the kernel of a Gaussian Process. Our idea is to create a continuous kernel space in the latent space of a variational autoencoder, and run an auxiliary optimization to identify the best kernel. Results show that the proposed method, Kernel Optimized Blackbox Optimization (KOBO), outperforms state of the art by estimating the optimal at considerably lower sample budgets. Results hold not only across synthetic benchmark functions but also in real applications. We show that a hearing aid may be personalized with fewer audio queries to the user, or a generative model could converge to desirable images from limited user ratings.

Figures

Figures reproduced from arXiv: 2507.20533 by the authors.

**Figure 1.** KerGPR in VAE latent space gives K∗ to fGPR the rough shape of f(x), i.e., select a GPR kernel that models this shape. With humans, for example, f(x) may have a staircase structure as they may not perceive differences in certain neighborhoods of x, but their ratings may change just outside that neighborhood. If GPR’s surrogate model ˆf(x) captures this staircase structure in its kernel, sample efficiency can improv… view at source ↗

**Figure 2.** Row 1 shows simple (or base) kernels (Duvenaud [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** connects all the modules to give a complete overview of KOBO. The main objective function (Eqn. 1) is optimized with a Function GPR (fGPR). fGPR uses a simple Square-Exponential (SE) kernel to obtain a batch of observations Dn. The model evidence is then passed to the kernel learning pipeline. The Kernel Combiner takes simple kernels and observations Dn as inputs, and outputs a discrete space of composite kernels K… view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: Comparison of KOBO and conventional BO using SE, PER, RQ, and Matern kernels (Bottom Row) for (a) Staircase, ´ (b) Smooth Branin, and, (c) Periodic Michalewicz functions (Top Row). the smooth shape. When real world functions exhibit complex (non-smooth) structures and…

**Figure 5.** Figure 5: Comparison of KOBO, MCMC, CKS, and BOMS for Staircase (Col 1), Branin (Col 2), and Michalewiez (Col 3) functions: (Row 1) Regret (Row 2) Model Evidence. 2000). These synthetic objective functions were sampled from a GP that uses different but known kernels. The subseq…

**Figure 6.** Figure 6: The blue curve is real-world CO2 emissions data from (Thoning, Tans, and Komhyr 1989). The red curve is KOBO’s prediction of the blue curve after observing (a) 20% (b) 40% (c) 60% of the blue data. xstart xQ=5 xQ=15 xQ=25 Prompt: Office room with a desk, a blue chair, …

**Figure 7.** Figure 7: Comparison of KOBO with rc lengths L = 5, 15, 25: (a) Regret (b) Model Evidence. Visualizing with 1D synthetic functions: To visualize kernel learning (as previously done for CO2 emissions data in Section 4 in main paper), we sample 1D functions from GPs employing dif…

**Figure 8.** Figure 8: Function structure modeling with KOBO: Objective function drawn from a GPR posterior using (a) Rational Quadratic (b) Matern, and (c) Periodic kernels. The black line denotes the true function. The orange line denotes the ´ fGPR posterior mean function model using Q = …

**Figure 9.** Figure 9: KerVAE reconstruction: (Block 1) Input, (Block 2) [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]

**Figure 10.** Figure 10: Comparison of KOBO, MCMC, CKS, and BOMS for Styblinski-Tang (Col 1), and Hartmann-6 (Col 2) functions: (Row 1) Regret (Row 2) Model Evidence. the user satisfaction f(h) is maximized if users are willing to listen and rate some audio clips (Q queries) prescribed by KO…

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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Anal- ysis of the NOAA GMCC data, 1974–1985.Journal of Geo- physical Research: Atmospheres, 94(D6): 8549–8565. Wang, J

work page 1974

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arXiv preprint arXiv:2009.10862

An intuitive tutorial to Gaussian processes regression. arXiv preprint arXiv:2009.10862. Wang, Z.; Gehring, C.; Kohli, P.; and Jegelka, S

arXiv 2009

[3] [10]

low effective dimensions

are employed for black-box optimization because they pro- vide a non-parametric mechanism to generate a probabilistic surrogate for the unknown functionf. Given a set of samples X ={x1,x 2,...,x K} at which the functionf has been ob- served, i.e., we knowF ={f (x1),f (x2),...,f (xK)}, we can identify an infinite number of candidate functions that match th...

work page 2018

[4] [11]

A note on KerGPR’s kernel: We do not learn KerGPR’s kernel – its objective, Model Evidence (Eqn

- An SE kernel is used in KerGPR, which runs in the kernel latent spaceZ of KerV AE. A note on KerGPR’s kernel: We do not learn KerGPR’s kernel – its objective, Model Evidence (Eqn. 4 in main paper) is sufficiently smooth, unlike the objec- tivef (x) offGPR whose discontinuous structure bene- fits greatly from kernel learning. Kernel learning is use- ful ...

work page 2017

[5] [12]

or Metropolis-Hastings (Gardner et al. 2017). To apply Metropolis-Hastings, the proposal distribution g(k′|k) is defined as follows; given a current kernel model k, we can either add or multiply a chosen base kernel from B. We construct the proposal distribution by first choosing whether to add or multiply, each with 50% probability, and next, picking a b...

work page 2017

[6] [14]

Table 4: Comparing grammar vs. one-hot encoding kC1 =A2∗B+D Grammar-based [2,1,0,0,0,0,0,0,1,0,0,0,0,0,0] One-hot [00001,010,00001,010,00010,001,01000,000,00000] kC2 =A2∗B+C∗D Grammar-based [2,1,0,0,0,0,0,1,1,0,0,0,0,0,0] One-hot [00001,010,00001,010,00010,001,00100,010,01000] With and without data-based representation: Data-based representationrd encodes...

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G.; and Rardin, R

Parker, R. G.; and Rardin, R. L. 2014.Discrete optimization. Elsevier. Rasmussen, C. E.; Williams, C. K.; et al

work page 2014

[8] [2011]

if a user rates few pictures generated by the model, can the model find the “best

database as the corrupting filter b1. Second, to emulate cheap speakers, we generate random dis- tortions by creating a random corrupting filterb2, eachb2[j] selected independently from [−30, 30]dB. A sample speech clipa is filtered with the distorting filter b1 orb2 to obtain the corrupted clip,r =b1:2∗a. The goal of the audio personalization task is to ...

work page 2003