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Exponential concentration in quantum kernel methods

Supanut Thanasilp, Samson Wang, M. Cerezo, Zoë Holmes

arXiv:2208.11060v2quant-phcs.LGstat.ML

TL;DR

Quantum kernel methods offer convex training, but their practical value depends on efficiently estimating kernels. This paper derives concentration bounds for four mechanisms, shows polynomial-shot models can become input-independent, and evaluates the effects numerically.

  • Problem

    Quantum kernel methods assume that hardware can estimate kernel values accurately and efficiently, but exponential concentration can undermine this assumption.

  • Method

    The paper analytically derives concentration bounds for expressivity, global measurements, entanglement, and noise, and studies parametrized embeddings with numerical simulations.

  • Results

    With polynomially many measurement shots, exponentially concentrated kernels can produce models whose predictions on unseen inputs are independent of the input data.

  • Takeaways & Limitations

    Problem-inspired embeddings and careful choices avoiding highly expressive, entangling, global-measurement, or noisy settings are indicated for efficient kernel evaluation.

Abstract

from arXiv · show

Kernel methods in Quantum Machine Learning (QML) have recently gained significant attention as a potential candidate for achieving a quantum advantage in data analysis. Among other attractive properties, when training a kernel-based model one is guaranteed to find the optimal model's parameters due to the convexity of the training landscape. However, this is based on the assumption that the quantum kernel can be efficiently obtained from quantum hardware. In this work we study the performance of quantum kernel models from the perspective of the resources needed to accurately estimate kernel values. We show that, under certain conditions, values of quantum kernels over different input data can be exponentially concentrated (in the number of qubits) towards some fixed value. Thus on training with a polynomial number of measurements, one ends up with a trivial model where the predictions on unseen inputs are independent of the input data. We identify four sources that can lead to concentration including: expressivity of data embedding, global measurements, entanglement and noise. For each source, an associated concentration bound of quantum kernels is analytically derived. Lastly, we show that when dealing with classical data, training a parametrized data embedding with a kernel alignment method is also susceptible to exponential concentration. Our results are verified through numerical simulations for several QML tasks. Altogether, we provide guidelines indicating that certain features should be avoided to ensure the efficient evaluation of quantum kernels and so the performance of quantum kernel methods.

I. INTRODUCTION

Quantum kernel methods combine convex training with potentially difficult kernel estimation. The paper shows that exponential concentration can make polynomial-shot models input-independent and identifies mechanisms and embedding guidelines relevant to avoiding it.

  • Quantum kernel methods are viewed as promising for practical quantum advantage because embeddings can target classically intractable patterns.
  • Convex loss landscapes guarantee optimal kernel-model parameters, but only when kernel values can be efficiently estimated to sufficient precision.
  • Under certain conditions, quantum kernels exponentially concentrate around a fixed value, requiring exponentially many measurement shots to resolve them.
  • With polynomially many measurements, the trained model can become independent of input data and perform trivially on unseen inputs.
  • The paper derives concentration bounds for expressivity, entanglement, global measurements, and noise, and supports the results with numerical learning-task experiments.
  • The authors suggest problem-inspired embeddings over highly expressive, entangling problem-agnostic embeddings, while warning that training parametrized embeddings can itself exhibit barren plateaus.

A. Framework

The framework embeds data into n-qubit quantum states and evaluates similarities through quantum kernels. Kernel models use the resulting Gram matrix in a convex optimization problem, provided that matrix can be obtained accurately and efficiently.

  • Training and evaluation: Training minimizes empirical loss on labeled data, after which the model is evaluated on unseen inputs; solving can require O(poly(Ns)) iterations under the Gram-matrix assumption.
  • Generalization: The framework asks whether successful training on a sufficiently large dataset yields accurate predictions on unseen data.
  • Framework: Quantum kernel learning encodes each input vector as an n-qubit state using an arbitrary unitary data embedding.
  • Quantum kernels: A quantum kernel is a similarity measure between two encoded states, with fidelity and projected kernels treated as common choices.
  • Quantum kernels: Projected kernels use reduced single-qubit states, the Schatten 2-norm, and a positive hyperparameter γ.
  • Kernel models: Kernel methods map data into a 2^n-dimensional Hilbert space, where inner products support models such as support vector machines.
  • Kernel models: The optimal kernel-model parameters are obtained from the Gram matrix through a convex optimization problem when the matrix is accurately available.

B. Why exponential concentration is problematic

Exponential concentration makes statistically estimated quantum kernels effectively independent of input data when only polynomially many shots are available. Training can remain convex and achieve low error, but predictions on unseen data become trivial and additional training data cannot resolve the problem.

  • Kernel estimation: Exponential concentration makes kernel values increasingly difficult to distinguish, requiring exponentially many shots for sufficiently accurate estimation.For bounded quantum kernels, an exponentially small mean directly implies exponentially small variance and exponentially unlikely constant deviations.
  • Kernel estimation: With polynomially many shots, estimated Gram matrices become independent of input data for both Loschmidt Echo and SWAP tests.The Loschmidt Echo estimate becomes the identity matrix with exponentially high probability, while the SWAP estimate becomes statistically indistinguishable from a data-independent random matrix.
  • Training versus generalization: The resulting failure is a generalization barrier rather than a training barrier: convex optimization still reaches the optimum and can achieve low training error.Training labels are effectively hard-coded through optimization, even though the learned model is insensitive to input data.
  • Generalization: Training on these estimates produces predictions on unseen inputs that are independent of the training data, regardless of the measurement strategy.Loschmidt Echo predictions are zero with exponentially high probability, whereas SWAP-test predictions follow data-independent random fluctuations.
  • Training versus generalization: Increasing the number of training points does not resolve this barrier, whereas numerical simulations reproduce the predicted lack of generalization improvement.For Loschmidt Echo, η(Ns) = 1 for all Ns; SWAP behavior matches training on a data-independent random matrix, despite zero training error.
  • Projected kernels: Projected quantum kernels require additional statistical estimation of reduced-state norms, using either single-qubit tomography or local SWAP tests.Their concentration analysis is therefore more involved than for the fidelity kernel.

C. Sources of exponential concentration

The paper identifies four features that can severely hinder information extraction through quantum kernels: expressive embeddings, entanglement, global measurements, and noise. It derives a concentration bound for each source.

  • Overview: The analysis examines when quantum kernel values concentrate because extracting useful information from quantum states becomes difficult.This investigation is motivated by the trivial data-independent models produced by exponential concentration.
  • Overview: Four sources of exponential concentration are identified: expressive data embeddings, entanglement, global measurements, and noise.The paper derives an associated concentration bound for each source.

1. Expressivity-induced concentration

The paper connects higher data-embedding expressivity with stronger concentration of fidelity and projected quantum kernels. Numerical results show that sufficiently expressive HEE embeddings produce exponentially decaying kernel variances, motivating data-structure-aware embeddings.

  • Expressivity measure: Expressivity measures how closely the data-encoded unitary ensemble covers the full unitary group, with zero distance indicating maximal expressivity.The measure incorporates both embedding expressivity and randomness of the input dataset.
  • Mechanism: Highly expressive embeddings make fidelity-kernel inputs resemble approximately random, nearly orthogonal vectors, causing typical kernel values to become exponentially small.The mechanism follows from comparing states in an exponentially large Hilbert space.
  • Analytical result: In the 2-design limit, fidelity-kernel values exponentially concentrate around the mean 1/2^n, requiring exponentially many measurement shots for evaluation.The theorem applies broadly to embedding architectures and also extends to quantum data.
  • Numerical verification: The MNIST experiment encodes PCA-reduced digit images with an L-layer HEE and evaluates kernel-value variance across distinct input pairs.Identical-input pairs, whose kernel values are always 1, are excluded from the variance calculation.
  • Numerical verification: Increasing HEE depth increases expressivity and eventually produces exponential concentration for both kernels; the projected kernel reaches this regime at roughly L ≥ 16, versus L ≥ 75 for the fidelity kernel.The fidelity kernel generally exhibits smaller variances and therefore stronger concentration.
  • Implication: The results indicate that highly expressive or unstructured data embeddings should generally be avoided in favor of embeddings that incorporate data structure.The paper gives geometrically inspired embeddings as an example of a problem-informed alternative.

2. Entanglement-induced concentration

Projected quantum kernels can concentrate because entanglement makes local reduced states approach maximally mixed states. Volume-law entanglement yields deterministic exponential concentration, whereas area-law scaling leaves the outcome unresolved by the bound.

  • Mechanism: For projected kernels, entanglement is a concentration source arising from non-local correlations between qubits after reducing the encoded states.This issue differs from the high-expressivity mechanism because projected kernels reduce feature-space dimension but retain local-state correlations.
  • Analytical result: Theorem 2 bounds deviation from the fixed value 1 using the relative entropy between each reduced encoded state and a single-qubit maximally mixed state.The bound is deterministic for a given pair of data-encoded states.
  • Volume-law regime: Volume-law entanglement with relative entropy in O(1/2^(n−1)) makes projected-kernel values deterministically concentrate exponentially to 1.The concentration can occur even when the embedding is not highly expressive.
  • Area-law regime: For area-law scaling with relative entropy in O(1), the theorem permits but does not guarantee avoidance of exponential concentration.The upper-bound nature of the result leaves this regime unresolved.
  • Learning consequence: With volume-law encoded training states, supervised learning using projected kernels fails with a polynomial number of measurement shots.This follows because the concentrated kernel values do not provide sufficient distinctions for the learning task.
  • Scope: Fidelity kernels with pure input states are not affected by this entanglement mechanism because they do not require tracing out qubits.The stated entanglement-induced bound concerns projected kernels.

3. Global-measurement-induced concentration

Global measurements can cause quantum-kernel values to concentrate because they extract information globally from an exponentially large Hilbert space. This effect depends on the embedding and data structure, and is not unavoidable.

  • Mechanism: Global measurements act non-trivially on all n qubits and are required for fidelity kernels but not projected kernels.Their global nature can make extracting useful state information difficult.
  • Analytical setting: Product-state embeddings with independently uniform input components can exhibit exponentially small fidelity overlaps under global measurement.The concentration follows intuitively because fidelities between product states are usually exponentially small.
  • Numerical verification: An embedding with a single layer of one-qubit rotations shows exponential decay in fidelity-kernel variance for the hypercube classification task.The simulation uses input components uniformly sampled from [−π, π].
  • Embedding expressivity: Increasing the expressivity of hardware-efficient embeddings can accelerate the exponential decay of kernel variance.Figure 7 compares one-qubit rotation embeddings with L-layer hardware-efficient embeddings.
  • Boundary conditions: Global measurements do not always cause exponential concentration when the data structure keeps fidelity-kernel values from concentrating worse than polynomially in n.MNIST with low-depth ansätze is given as an example where the proposition’s assumptions fail.
  • Design guidance: For global-kernel evaluation, encoded-state fidelities should remain at least in Ω(1/poly(n)), using problem structure or reduced embedding expressivity.This is the paper’s design guidance for avoiding severe concentration.

4. Noise-induced concentration

Hardware noise drives encoded quantum states toward the maximally mixed state, producing concentration of both fidelity and projected quantum kernels. The effect strengthens with circuit depth and noise, creating a scalability concern for deep embeddings.

  • Noise model: The noise model applies local Pauli channels before and after every layer of an L-layer data-embedding circuit.The framework covers hardware-efficient, QAOA, Hamiltonian variational, and IQP embeddings.
  • Analytical result: Noise-induced concentration is exponential in the number of layers L for both fidelity and projected quantum kernels.The bounds are deterministic because the noise acts independently of the input data.
  • Fixed points: The noisy data-encoded state concentrates toward the maximally mixed state.For fidelity kernels the fixed point is µ = 1/2^n, while for projected kernels it is µ = 1.
  • Scalability: When L scales polynomially with n, the bounds indicate exponential kernel concentration from noise in the regime of moderately deep circuits.This regime is relevant for embeddings with local connectivity intended to resist classical simulation.
  • Numerical verification: On MNIST, kernel values show exponential concentration with depth L, and concentration is stronger for larger noise parameter q.Figure 8 evaluates fidelity and projected kernels at n = 8 with Ns = 40.
  • Practical limitation: The paper argues that common error-mitigation techniques cannot resolve noise-induced exponential concentration.The listed techniques include zero-noise extrapolation, Clifford data regression, virtual distillation, and probabilistic error cancellation.

D. Training parameterized quantum kernels

Training a parameterized data embedding with kernel target alignment can itself suffer exponential concentration. Under the same kinds of embedding features that produce barren plateaus, the alignment landscape becomes exponentially flat and untrainable with polynomially many shots.

  • Setup: Parameterized embeddings U(x, θ) generate a family of kernels κθ(x, x′) whose parameters can be optimized before the standard kernel algorithm.The parameters typically correspond to single-qubit rotation angles.
  • Training objective: Kernel target alignment measures similarity between a parameterized kernel and an approximated ideal kernel derived from training labels.For binary classification, the ideal kernel is positive for same-class pairs and negative for different-class pairs.
  • Optimization: Unlike model training, embedding optimization with target alignment generally has a non-convex loss landscape.Target alignment can nevertheless be used as the loss function for aligning the parameterized kernel with the ideal kernel.
  • Concentration bound: The variance of target alignment is upper bounded by the variances of the parameterized quantum kernels.This connects kernel-value concentration directly to the trainability of the embedding objective.
  • Trainability consequence: If all training-pair kernel variances vanish exponentially with qubit number, target alignment becomes exponentially flat and untrainable with polynomially many measurement shots.The deviation probability from the mean also vanishes exponentially with problem size.
  • Design implications: Deep unstructured circuits and global measurements, which are detrimental for QNN trainability, likewise produce barren plateaus for parameterized quantum kernels.These features should therefore be avoided when designing trainable data embeddings.

III. DISCUSSION

The discussion places exponential concentration alongside other QML scalability barriers and clarifies that kernel convexity is useful only when kernel values can be estimated efficiently. It recommends embeddings that preserve data-dependent distinctions while recognizing important scope conditions and prior work.

  • Implications: Kernel-method trainability guarantees require quantum-kernel values to be estimated with sufficient precision to retain information about input data.Convexity alone does not ensure a useful trained model.
  • Implications: With polynomially many shots, exponentially concentrated kernels can yield predictions independent of unseen inputs and trivial generalization.Increasing the number of training data points does not fix this; more shots or a more appropriate embedding are needed.
  • Design guidance: The paper identifies expressivity, global measurements, entanglement, and noise as four aspects requiring care when choosing quantum-kernel embeddings.Problem-agnostic embeddings are described as typically highly expressive and generally best avoided.
  • Parameterized embeddings: Training parameterized quantum kernels with kernel target alignment also suffers an exponentially flat landscape under similar conditions.Thus embedding optimization introduces a trainability issue in addition to the standard kernel-model training process.
  • Relation to prior work: Earlier fidelity-kernel studies highlighted exponential-concentration problems but did not analyze their exact causes in detail.Those studies also assumed direct access to quantum states without shot noise.
  • Projected kernels: Prior work showed that projected kernels can maintain good generalization where fidelity kernels fail, while their evaluation may still traverse the exponentially large Hilbert space.The paper examines expressivity and concentration questions left open for projected kernels.
  • Mitigation attempts: A quantum Fisher kernel was analytically shown to avoid exponential concentration with log-depth local layers, whereas linear depth can restore high expressivity and concentration.This comparison illustrates that depth remains a relevant scope condition.

Appendix C: Practical implications of exponential concentration on kernel methods

With polynomially many measurements, exponential concentration can make estimated quantum-kernel matrices and model outputs independent of the input data. The section considers fidelity and projected kernels, their measurement strategies, and the resulting resolution requirements.

  • Practical consequences: Polynomial-shot estimates can produce a Gram matrix and trained model that are independent of input data, preventing generalization.The consequence is established for kernel ridge regression and is stated to carry over to other learning tasks.
  • Measurement strategies: The fidelity quantum kernel is analyzed using Loschmidt Echo and SWAP tests, while the projected quantum kernel is studied with two measurement strategies.The Loschmidt Echo uses all-zero outcomes, whereas the SWAP test uses an ancilla Pauli Z expectation.
  • Fidelity quantum kernel: In the Loschmidt Echo test, kernel estimates concentrate at zero, making the estimated Gram matrix likely to be the identity and predictions likely to be zero.Numerical simulations are provided to support the fidelity-kernel theoretical results.
  • Statistical indistinguishability: For the SWAP and projected-kernel strategies, exponentially concentrated kernel values become statistically indistinguishable, as do the resulting model predictions.The projected-kernel section also includes numerical results supporting the theoretical findings.
  • Estimation framework: Kernel values are estimated from measurement outcomes of an observable, with the estimate formed from N measurement shots and outcome probabilities determined by the quantum state.For fidelity kernels, the observable and state depend on the selected measurement strategy.
  • Definition: Exponential concentration is defined as concentration toward an input-independent value, with exponentially small deviation probability; the limiting value may itself be exponentially small.The definition distinguishes deterministic and probabilistic concentration.

a. Loschmidt Echo test

The Loschmidt Echo test estimates fidelity kernels by counting all-zero bitstrings. When the kernel concentrates toward an exponentially small value, polynomially many shots yield zero estimates, an identity Gram matrix, and input-insensitive predictions with exponentially high probability.

  • Measurement: The Loschmidt Echo estimates the fidelity kernel as the empirical probability of measuring the all-zero bitstring.Its observable is a global projector onto the all-zero state, with outcomes +1 for all-zero and 0 otherwise.
  • Kernel estimates: With polynomially many shots, exponentially concentrated fidelity kernels are estimated as zero with probability exponentially close to 1.The result assumes concentration toward an exponentially small value.
  • Gram matrix: For a polynomial-size training set, the estimated Gram matrix equals the identity with probability exponentially close to 1.The argument combines zero off-diagonal kernel estimates across the training pairs.
  • Generalization: Predictions on unseen inputs concentrate at zero with high probability because all estimated kernels connecting them to the training data are zero.The model therefore remains insensitive to input data despite potentially low training error.
  • Training: Kernel ridge regression then has input-data-independent optimal parameters with probability at least 1 − δ, where δ ∈ O(c−n) for some c > 1.The fixed point is a0(y, λ) = y/(1 − λ).
  • Practical boundary: Good generalization cannot be restored simply by adding training points; at least an exponential number of shots is required for hope of resolving the kernels.The limitation concerns measurement resolution rather than the convex optimization landscape.

b. SWAP test

With polynomially many shots, exponentially concentrated kernel values become statistically indistinguishable from data-independent distributions. Consequently, the estimated Gram matrix and trained model lose dependence on the input data, despite potentially low training error.

  • Statistical indistinguishability: Polynomially many samples cannot distinguish exponentially concentrated fidelity-kernel measurements from a uniform binary distribution with exponentially high probability.The kernel value lies within an exponentially small interval around its concentration value for almost all input pairs.
  • Statistical indistinguishability: Post-processing polynomial-shot kernel estimates produces models whose predictions are independent of the input data for supervised and unsupervised tasks.The resulting models are statistically indistinguishable from models built from the uniform distribution.
  • Fidelity-kernel ridge regression: In fidelity-kernel ridge regression, the estimated Gram matrix, optimal parameters, and unseen-input predictions are each statistically indistinguishable from input-data-independent random quantities.A polynomial-size training set preserves exponentially high probability for the joint indistinguishability of the estimated kernel entries.
  • Fidelity-kernel ridge regression: The model may retain low training error while generalizing poorly because labels are effectively incorporated into a model that is insensitive to its inputs.Good generalization requires data-dependent outputs, whereas concentrated estimates can still fit the training labels.
  • Projected quantum kernels: The projected-kernel concentration mechanism drives the underlying local SWAP measurement distribution toward the fixed distribution {3/4, 1/4}.For tomography, individual reduced-state coefficients instead become indistinguishable from the uniform distribution {1/2, 1/2}.
  • Projected quantum kernels: For projected quantum kernels, polynomial-shot tomography and local SWAP estimates of reduced-state distances become input-data-independent random variables with exponentially high probability.The corresponding kernel estimates and Gram matrices inherit this indistinguishability, preventing useful downstream kernel-model behavior.

Appendix D: Proof of Theorem 1: Expressivity-induced concentration

Theorem 1 bounds expressivity-induced concentration for fidelity and projected quantum kernels, using variance estimates and Chebyshev’s inequality. The result extends to different input distributions with modified expressivity-dependent bounds.

  • Theorem 1: Theorem 1 relates data-encoded-unitary expressivity to concentration of fidelity and projected quantum kernels.It assumes x and x′ are drawn from the same distribution and derives bounds for both kernel types.
  • Projected kernel: The projected-kernel bound uses ˜βHaar = 3/(2^n+1)+2.The projected-kernel proof separately bounds the reduced-state terms before applying concentration arguments.
  • Proof strategy: The proof upper-bounds each kernel’s variance and then applies Chebyshev’s inequality to obtain concentration.For the fidelity kernel, the proof explicitly proceeds through variance control before invoking Chebyshev’s inequality.
  • Fidelity kernel: The fidelity-kernel concentration point is βHaar = 1/[2^(n−1)(2^n+1)].This value is obtained by explicitly performing Haar integration for pure input states.
  • Different input distributions: For inputs from different distributions, the same concentration form holds with bounds depending on εUx and εUx′.The fidelity bound uses βHaar + εUxεUx′ + √βHaar(εUx + εUx′), while the projected bound uses 2γn(2˜βHaar + εUx + εUx′).

Appendix F: Proof of Proposition 3: Global-measurement-induced concentration

The appendix analyzes global-measurement-induced concentration for product embeddings whose input components are independently sampled. In the fully random single-qubit limit, the fidelity kernel concentrates exponentially.

  • Proposition 3: Proposition 3 studies a fidelity kernel with a product embedding of independent uniformly sampled input components and single-qubit y-axis rotations.The initial state is assumed to be a product state.
  • Proof strategy: The proof bounds the variance over input data and applies Chebyshev’s inequality to derive the concentration result.Independence of input components factorizes the relevant expressions.
  • Concentration point: The concentration point becomes exponentially small as the number of qubits increases.For the specified y-axis rotations, each factor in the product evaluates to 1/2.
  • Generalization: The generalized proposition replaces the specific rotations with general single-qubit unitaries and uses local expressivity measures εUk.The product-state and independent-input assumptions remain part of the generalized setting.
  • Random-unitary limit: When every single-qubit unitary is random, the variance bound is 1/3^n and the kernel concentrates probabilistically.This is the limit εUk = 0 for every qubit.

Appendix G: Proof of Theorem 3: Noise-induced concentration

Theorem 3 establishes noise-induced concentration for fidelity and projected quantum kernels under layered local Pauli noise. The noisy encoded states themselves converge toward the maximally mixed state.

  • Proof strategy: The proof uses Pauli decompositions, unitary invariance, noise-channel contraction, and entropy-based inequalities.The kernel bounds follow after repeatedly applying these tools across circuit layers.
  • Theorem 3: Theorem 3 considers an L-layer data-embedding circuit with layer-wise Pauli noise characterized by q < 1.It provides concentration bounds for both fidelity and projected noisy kernels.
  • Fidelity kernel: For the noisy fidelity kernel, the concentration point is μ = 1/2^n.The theorem states this value explicitly for ˜κFQ(x, x′).
  • Projected kernel: For the noisy projected kernel, the concentration point is μ = 1.The theorem states this value explicitly for ˜κPQ(x, x′).
  • State concentration: The noisy data-encoded state concentrates toward the maximally mixed state.The proof tracks Pauli coefficients through unitary transformations and local noise channels.

Appendix H: Error Mitigation

The appendix examines whether error mitigation can prevent concentration and extends the analysis to kernel-target-alignment training landscapes. It concludes that standard mitigation cannot generally remove noise-induced data-independence under the stated depth conditions.

  • Error mitigation: Error-mitigation protocols can suppress expectation-value errors yet still leave exponentially flat landscapes or require exponential resources.This motivates testing whether mitigation resolves concentration in quantum kernels.
  • Mitigated observables: For circuits with local depolarizing noise and depth Lσ ∈ Ω(n), mitigated expectation values concentrate toward a fixed point F0.The result assumes polynomially bounded observable norm and bounded mitigation coefficients and copy number.
  • Kernel estimation: Quantum-kernel estimators are expectation values, so the general error-mitigation concentration theorem applies to them.The fidelity kernel can use a projection measurement or a SWAP test, while projected kernels use local Pauli measurements or SWAP tests.
  • Limitation of mitigation: Standard error-mitigation strategies cannot generally mitigate noise-induced exponential concentration of kernel values.The appendix does not rule out improvements for small circuit depth L.
  • Kernel alignment: Kernel-target-alignment concentration can be bounded through concentration of the parameterized quantum kernels.If the kernel landscape is exponentially flat in n, the target-alignment landscape suffers the same behavior.
  • Parameterized embeddings: Entanglement in parameterized data embeddings can negatively affect projected quantum kernels, while barren plateaus in kernel components imply fidelity-kernel concentration.These results connect parameterized-kernel training landscapes to concentration mechanisms.
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