Spectral Truncation Regularization

• Spectral truncation regularization is a technique that applies a hard cutoff to retain only the most stable spectral components, thereby controlling noise amplification and bias-variance trade-offs in inverse problems.
• It is implemented in various settings—including truncated SVD, kernel ridge regression, and deep network regularization—to achieve efficient, stable inversion by excluding noisy or computationally burdensome spectral directions.
• Recent advances leverage weak convergence theory and sampling-inequality analysis to optimize truncation levels, enhancing both theoretical insights and the practical performance across different applications.

Spectral truncation regularization denotes a family of regularization procedures in which inversion, estimation, or representation is restricted to a selected spectral subspace while the remaining spectral components are suppressed. In the classical inverse-problems sense, it is the spectral cutoff or truncated SVD rule

$$x_k=\sum_{i=1}^k \frac{\langle y^\delta,u_i\rangle_{\mathcal H_2}}{\sigma_i}\,v_i,$$

equivalently the filter

$s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$

which inverts only the retained singular band and discards the rest (Guastavino et al., 4 Dec 2025). The same structural idea appears in spectrally truncated kernel ridge regression, FPCA-based functional autoregression, noncommutative kernel constructions based on Fourier projections, frequency-domain activation pruning, and Liouvillian mode truncation for open-system OTOCs (Amini, 2019, Zhao, 28 Apr 2026, Hashimoto et al., 2024, Khan et al., 2017, Bergamasco et al., 5 Mar 2025). Across these settings, the central regularizing mechanism is identical: unstable, noisy, weak, or computationally expensive spectral directions are excluded from the effective model.

1. Canonical spectral formulation

The operator-theoretic prototype starts from a compact linear operator $A:\mathcal H_1\to\mathcal H_2$ with singular system $\{(\sigma_j,u_j,v_j)\}_j$, so that

$$A v = \sum_{j=1}^\infty \sigma_j \langle v_j, v\rangle_{\mathcal{H}_1} u_j, \qquad A^* u = \sum_{j=1}^\infty \sigma_j \langle u_j, u\rangle_{\mathcal{H}_2} v_j.$$

Within the general spectral-regularization framework,

$\mathfrak{R}(B,z,\lambda):=s_\lambda(B^*B)\,B^*z,$

spectral truncation is the special case in which $s_\lambda$ is the hard cutoff above. In index form, choosing $\lambda=\sigma_k^2$ makes the threshold formulation equivalent to retaining the first $k$ singular components (Guastavino et al., 4 Dec 2025).

The broader spectral-filter viewpoint treats truncation as one member of a larger family. In the spectral reconstruction-operator framework, a linear spectral estimator has the form

$R(y;g)=\sum_n g_n \langle y,v_n\rangle u_n,$

with truncated SVD given by $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$0 for $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$1 and $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$2 otherwise. This places hard cutoff alongside smooth filters such as Tikhonov and Landweber iteration (Burger et al., 2023).

Method	Spectral coefficient/filter	Effect
Spectral truncation / TSVD	$s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$3 for $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$4, else $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$5	Hard cutoff
Tikhonov	$s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$6	Soft shrinkage
Landweber	$s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$7	Iterative spectral damping

What distinguishes truncation is not merely spectral dependence, but discontinuous selection. The retained modes are inverted exactly; all others are set to zero. This produces a particularly transparent bias–variance decomposition and makes truncation the reference model for many later generalizations.

2. Weak convergence theory and sampling-inequality analysis

A central recent development is the analysis of spectral truncation inside a source-condition-free theory based on sampling inequalities. For the semidiscrete estimator

$s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$8

the paper on weak convergence rates generalizes classical sampling inequalities from Tikhonov regularization to arbitrary spectral filters satisfying Assumption 1, then transfers those estimates from the direct RKHS variable $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$9 back to the inverse variable $A:\mathcal H_1\to\mathcal H_2$0 in a weak sense (Guastavino et al., 4 Dec 2025).

The decisive shift is methodological. Classical strong convergence rates in $A:\mathcal H_1\to\mathcal H_2$1 normally require source conditions such as $A:\mathcal H_1\to\mathcal H_2$2. By contrast, the sampling-inequality approach works in the RKHS $A:\mathcal H_1\to\mathcal H_2$3, controls the approximation error through fill distance $A:\mathcal H_1\to\mathcal H_2$4, kernel smoothness, and filter-dependent terms such as $A:\mathcal H_1\to\mathcal H_2$5, and then evaluates the inverse error only against test functionals depending on $A:\mathcal H_1\to\mathcal H_2$6. Two classes are used: $A:\mathcal H_1\to\mathcal H_2$7, with

$A:\mathcal H_1\to\mathcal H_2$8

and $A:\mathcal H_1\to\mathcal H_2$9, with

$\{(\sigma_j,u_j,v_j)\}_j$0

For quasi-uniform sampling, where $\{(\sigma_j,u_j,v_j)\}_j$1, the resulting weak rates are explicit. In the uniformly bounded case,

$\{(\sigma_j,u_j,v_j)\}_j$2

and if $\{(\sigma_j,u_j,v_j)\}_j$3 is trace class,

$\{(\sigma_j,u_j,v_j)\}_j$4

These estimates apply directly to spectral cutoff, for which $\{(\sigma_j,u_j,v_j)\}_j$5, $\{(\sigma_j,u_j,v_j)\}_j$6, and the qualification $\{(\sigma_j,u_j,v_j)\}_j$7 is arbitrary. The same analysis also makes explicit the trade-off that remains after source conditions are removed: the estimates are weak rather than strong, and their exponents saturate at variance-limited ceilings such as $\{(\sigma_j,u_j,v_j)\}_j$8 and $\{(\sigma_j,u_j,v_j)\}_j$9 (Guastavino et al., 4 Dec 2025).

3. Kernel, regression, and operator-learning variants

In kernel ridge regression, spectral truncation appears as rank-restricted inversion of the empirical kernel matrix. If $$A v = \sum_{j=1}^\infty \sigma_j \langle v_j, v\rangle_{\mathcal{H}_1} u_j, \qquad A^* u = \sum_{j=1}^\infty \sigma_j \langle u_j, u\rangle_{\mathcal{H}_2} v_j.$$0 and $$A v = \sum_{j=1}^\infty \sigma_j \langle v_j, v\rangle_{\mathcal{H}_1} u_j, \qquad A^* u = \sum_{j=1}^\infty \sigma_j \langle u_j, u\rangle_{\mathcal{H}_2} v_j.$$1 is the rank-$$A v = \sum_{j=1}^\infty \sigma_j \langle v_j, v\rangle_{\mathcal{H}_1} u_j, \qquad A^* u = \sum_{j=1}^\infty \sigma_j \langle u_j, u\rangle_{\mathcal{H}_2} v_j.$$2 truncation, the spectrally truncated KRR estimator is

$$A v = \sum_{j=1}^\infty \sigma_j \langle v_j, v\rangle_{\mathcal{H}_1} u_j, \qquad A^* u = \sum_{j=1}^\infty \sigma_j \langle u_j, u\rangle_{\mathcal{H}_2} v_j.$$3

Its exact minimax empirical $$A v = \sum_{j=1}^\infty \sigma_j \langle v_j, v\rangle_{\mathcal{H}_1} u_j, \qquad A^* u = \sum_{j=1}^\infty \sigma_j \langle u_j, u\rangle_{\mathcal{H}_2} v_j.$$4 risk over the RKHS unit ball is

$$A v = \sum_{j=1}^\infty \sigma_j \langle v_j, v\rangle_{\mathcal{H}_1} u_j, \qquad A^* u = \sum_{j=1}^\infty \sigma_j \langle u_j, u\rangle_{\mathcal{H}_2} v_j.$$5

This formula isolates the truncation bias through $$A v = \sum_{j=1}^\infty \sigma_j \langle v_j, v\rangle_{\mathcal{H}_1} u_j, \qquad A^* u = \sum_{j=1}^\infty \sigma_j \langle u_j, u\rangle_{\mathcal{H}_2} v_j.$$6 and the variance reduction through the truncated sum. The striking consequence is a finite-sample “free lunch”: for every $$A v = \sum_{j=1}^\infty \sigma_j \langle v_j, v\rangle_{\mathcal{H}_1} u_j, \qquad A^* u = \sum_{j=1}^\infty \sigma_j \langle u_j, u\rangle_{\mathcal{H}_2} v_j.$$7, spectrally truncated KRR, optimized over $$A v = \sum_{j=1}^\infty \sigma_j \langle v_j, v\rangle_{\mathcal{H}_1} u_j, \qquad A^* u = \sum_{j=1}^\infty \sigma_j \langle u_j, u\rangle_{\mathcal{H}_2} v_j.$$8, has minimax risk no larger than full KRR, and the inequality is strict whenever $$A v = \sum_{j=1}^\infty \sigma_j \langle v_j, v\rangle_{\mathcal{H}_1} u_j, \qquad A^* u = \sum_{j=1}^\infty \sigma_j \langle u_j, u\rangle_{\mathcal{H}_2} v_j.$$9 (Amini, 2019).

A different computational realization is the distribution-free TKRR estimator, which replaces the full normalized Gram matrix $\mathfrak{R}(B,z,\lambda):=s_\lambda(B^*B)\,B^*z,$0 by its main $\mathfrak{R}(B,z,\lambda):=s_\lambda(B^*B)\,B^*z,$1 submatrix

$\mathfrak{R}(B,z,\lambda):=s_\lambda(B^*B)\,B^*z,$2

and solves

$\mathfrak{R}(B,z,\lambda):=s_\lambda(B^*B)\,B^*z,$3

Its empirical-risk bound contains a variance term driven by the singular values $\mathfrak{R}(B,z,\lambda):=s_\lambda(B^*B)\,B^*z,$4 and bias terms governed by $\mathfrak{R}(B,z,\lambda):=s_\lambda(B^*B)\,B^*z,$5 and the spectral tail of the kernel operator. Under the paper’s exponential and polynomial decay assumptions, explicit choices of $\mathfrak{R}(B,z,\lambda):=s_\lambda(B^*B)\,B^*z,$6 and $\mathfrak{R}(B,z,\lambda):=s_\lambda(B^*B)\,B^*z,$7 yield the same optimal convergence rate as full KRR while reducing complexity from $\mathfrak{R}(B,z,\lambda):=s_\lambda(B^*B)\,B^*z,$8 to $\mathfrak{R}(B,z,\lambda):=s_\lambda(B^*B)\,B^*z,$9 (Saber et al., 2023).

Functional autoregression supplies a further instance in which truncation is structurally natural but statistically delicate. In FAR(1), the standard FPCA estimator uses the truncated pseudoinverse

$s_\lambda$0

with $s_\lambda$1 often selected by a cumulative-variance threshold such as $s_\lambda$2. The continuous-regularization study shows that this truncation choice is highly regime dependent: the optimal $s_\lambda$3 can differ by an order of magnitude across regimes, while $s_\lambda$4 and $s_\lambda$5 thresholds can inflate forecast error by up to $s_\lambda$6 relative to an oracle benchmark. The proposed Tikhonov alternative,

$s_\lambda$7

is explicitly presented as a replacement for discrete truncation rather than another form of truncation itself (Zhao, 28 Apr 2026).

Spectral truncation also enters kernel design at the level of the feature algebra. In the $s_\lambda$8-algebraic construction based on $s_\lambda$9, the projector

$\lambda=\sigma_k^2$0

implements a hard Fourier cutoff, and the truncated multiplication operator $\lambda=\sigma_k^2$1 is represented by the Toeplitz matrix $\lambda=\sigma_k^2$2. Kernels such as

$\lambda=\sigma_k^2$3

thereby interpolate between commutative pointwise kernels and finite-$\lambda=\sigma_k^2$4 noncommutative kernels. As $\lambda=\sigma_k^2$5, the Fejér reconstruction $\lambda=\sigma_k^2$6 converges pointwise to $\lambda=\sigma_k^2$7, and the truncation-induced noncommutativity disappears (Hashimoto et al., 2024).

4. Ill-posed evolution equations, spectroscopy, and open-system dynamics

In abstract evolution equations, spectral truncation regularization is often defined directly by spectral calculus. For the ill-posed parabolic final value problem, with $\lambda=\sigma_k^2$8 positive self-adjoint and spectral family $\lambda=\sigma_k^2$9, the truncated inverse is

$k$0

The cutoff removes the unstable high-frequency components $k$1 that would otherwise be exponentially amplified by $k$2. Under a general source condition $k$3 with $k$4, the total error obeys

$k$5

For exponential source conditions $k$6, the method yields a Hölder-type rate $k$7, and the paper emphasizes that, unlike Lavrentiev regularization in this setting, the truncation method has no index of saturation (Nair, 2019).

In inverse spectroscopy, truncation is tied to data-driven calibration. The spectroscopy study introduces an envelope

$k$8

where $k$9 is a lower spectral truncation of $R(y;g)=\sum_n g_n \langle y,v_n\rangle u_n,$0. Because $R(y;g)=\sum_n g_n \langle y,v_n\rangle u_n,$1 corresponds to a threshold on $R(y;g)=\sum_n g_n \langle y,v_n\rangle u_n,$2, it induces the truncation index

$R(y;g)=\sum_n g_n \langle y,v_n\rangle u_n,$3

The proposed “method of training examples” selects $R(y;g)=\sum_n g_n \langle y,v_n\rangle u_n,$4 by tangency between $R(y;g)=\sum_n g_n \langle y,v_n\rangle u_n,$5 and empirical relative-error curves from nearby model problems. In the numerical example, the chosen truncation parameter is $R(y;g)=\sum_n g_n \langle y,v_n\rangle u_n,$6, the selected Tikhonov parameter is $R(y;g)=\sum_n g_n \langle y,v_n\rangle u_n,$7, and the predicted relative error is $R(y;g)=\sum_n g_n \langle y,v_n\rangle u_n,$8 (Sizikov et al., 2015).

In dissipative quantum dynamics, spectral truncation regularization takes the form of modal reduction of the Liouvillian expansion of an OTOC. With right and left eigenvectors of the adjoint Liouvillian,

$R(y;g)=\sum_n g_n \langle y,v_n\rangle u_n,$9

the OTOC is expanded as

$s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$00

The truncation criterion retains the smallest set $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$01 for which the discarded tail over a target window remains below a tolerance: $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$02 The paper identifies two regimes: an intermediate-time regime $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$03 in which a small set of subdominant modes matters, and a long-time regime $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$04 dominated by the spectral gap $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$05. Broadly across parameters, fewer than $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$06 subdominant eigenvalues suffice for accurate reconstruction of the OTOC over the intermediate window (Bergamasco et al., 5 Mar 2025).

5. Neural-network and model-merging adaptations

In deep networks, the most literal truncation-based regularizer in the supplied literature is Spectral Dropout. After an intermediate convolutional block produces activations $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$07, the method applies a fixed orthogonal transform $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$08, masks spectral coefficients, and inverts: $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$09 The mask can be deterministic,

$s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$10

or can combine thresholding with Bernoulli gating. The paper explicitly interprets this as hard spectral truncation akin to truncated SVD or spectral shrinkage. Empirically, compared to Dropout and Drop-Connect, it speeds up convergence roughly $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$11, achieves an increase of $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$12 in pruning rate, and attains its best CIFAR-10 validation accuracy when about $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$13 of coefficients are pruned (Khan et al., 2017).

A distinct line of work uses spectral quantities without hard truncation. In continual learning, the objective

$s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$14

keeps the maximum singular value of each layer close to one. The paper is explicit that this method is not an explicit truncation or projection: it does not clip singular values and does not project onto $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$15. Rather, it is a soft spectral penalty motivated by gradient diversity, Jacobian conditioning, and Lipschitz control (Lewandowski et al., 2024). This distinction is central because much of the contemporary “spectral regularization” literature in deep learning is not truncation in the strict sense.

A hybrid formulation appears in data-free multi-task model merging. There the layerwise normal equation $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$16 is treated as a noisy inverse problem, and the closed-form SWUDI estimator applies a hard top-$s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$17 truncation together with the exponential filter associated with gradient flow: $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$18 The hard mask suppresses the noise-amplifying spectral tail, while the exponential factor reproduces the implicit regularization of early-stopped descent. Across the reported benchmarks, the resulting spectral solvers match or outperform state-of-the-art merging methods while reducing wall-clock time by $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$19–$s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$20 and peak GPU memory by up to $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$21 (Wei et al., 5 Jun 2026).

6. Parameter choice, trade-offs, and related misconceptions

The principal design problem in spectral truncation regularization is the choice of the truncation level itself: $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$22, $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$23, $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$24, $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$25, $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$26, $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$27, or an equivalent threshold $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$28. Across the surveyed literature, this choice is always a bias–variance or approximation–stability balance, but the controlling quantities differ. In classical inverse problems the balance is between approximation tails and amplified data noise; in sampling-based weak convergence it is between fill distance, kernel smoothness, and the variance term; in KRR it is between $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$29 and the truncated effective-dimension term; in FAR(1) it is strongly regime dependent; and in OTOC modeling it is tied to the target time window and modal weights.

One persistent misconception is that spectral truncation can replace all other regularization. The exact minimax analysis of spectrally truncated KRR rejects that interpretation: truncation alone is not a substitute for Hilbert-norm regularization, and the paper states that both are needed to achieve the best performance (Amini, 2019). A second misconception is that every spectral penalty is a form of truncation. In the spectral learning model for inverse problems, the supervised MSE-optimal learned filter is

$s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$30

which is Tikhonov-like and only mimics truncation when the ratio $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$31 exhibits a sharp spectral transition (Burger et al., 2023). Likewise, in combinatorial learning over $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$32, the $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$33 penalty on Walsh–Hadamard coefficients is a soft-thresholding mechanism. The paper shows that, under RSI or under QG plus approximate interpolation, it attains the same $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$34 statistical scaling as ideal hard $s_\lambda(t)= \begin{cases} \frac{1}{t}, & t\ge \lambda,\[4pt] 0, & t<\lambda, \end{cases}$35-sparse truncation, but it remains a convex sparsity surrogate rather than a hard cutoff (Aghazadeh et al., 2022).

The literature therefore supports a sharp conceptual separation. In the strict sense, spectral truncation regularization is a hard spectral projector followed by exact inversion or exact retention on the kept subspace. Around that core lie smooth spectral shrinkers, hybrid filters, and learned spectral operators that often approximate truncation, compete with it, or are introduced precisely because hard truncation is too brittle for a given regime. The modern theory does not eliminate truncation; instead, it clarifies when hard cutoff is optimal, when it needs to be supplemented by ridge-type damping, and when a continuous spectral filter is the more stable surrogate.