COS Procedure: Convex Oscillation Suppression

• COS procedure is a cross-disciplinary technique that combines convexification with oscillation suppression to stabilize numerical methods.
• In statistical learning, it uses blockwise comparisons and convex averaging to control empirical-process oscillations and recover near-convex performance.
• In PDE discretizations and non-convex optimization, COS manifests as convex constraints or inner-convex approximations, ensuring stability and reduced spurious oscillations.

Searching arXiv for the cited COS-related papers and terminology. arxiv_search(query="Convex-Oscillation-Suppressing OR COS Procedure oscillation suppression convex", max_results=10) arxiv_search(query="(Mendelson, 2017) OR (Wu, 31 Mar 2026) OR (Ding et al., 2024) OR (Virgili-Llop et al., 2018) OR (Wang et al., 2 Dec 2025)", max_results=10) Convex-Oscillation-Suppressing (COS) Procedure denotes, across several research literatures, a class of constructions that combine convexity or convexification with mechanisms intended to suppress undesirable oscillatory behavior. In statistical learning, the label has been used retrospectively for an unrestricted procedure that suppresses empirical-process oscillations through robust blockwise comparisons and exploits convex averaging to recover convex-case rates even for non-convex classes. In high-order finite volume and discontinuous Galerkin discretizations, it appears as an explicit oscillation-suppressing convex constraint enforced by radial scaling. In non-convex optimization, it describes sequential convex programming based on inner-convex majorizers that prevent feasibility oscillations and cost increases. In computational probability, however, COS refers to the Fourier–cosine expansion method, where convergence is governed by tail cosine energy rather than convex admissibility. This suggests that COS is best understood as a cross-domain technical motif rather than a single universally standardized algorithm (Mendelson, 2017, Wu, 31 Mar 2026, Virgili-Llop et al., 2018, Wang et al., 2 Dec 2025).

1. Terminological scope and principal usages

The expression “COS” does not have a single fixed meaning across arXiv-indexed research. In some works it is an explicit part of the formal framework; in others it is a reconstruction or interpretive lens applied to a method that is not originally named that way. A common source of confusion is therefore terminological rather than mathematical.

Domain	COS meaning	Representative source
Statistical learning	A reconstructed “Convex-Oscillation-Suppressing” view of an unrestricted learner	(Mendelson, 2017)
FV/DG stabilization	An explicit oscillation-suppressing convex constraint inside EPO	(Wu, 31 Mar 2026)
DG on triangular meshes	A reconstructed COS-like combination of RIOE damping and BP limiting	(Ding et al., 2024)
Non-convex optimization	A tailored COS interpretation of sequential convex programming with inner-convex surrogates	(Virgili-Llop et al., 2018)
Computational probability	The Fourier–cosine expansion method	(Wang et al., 2 Dec 2025)

In the learning paper, the procedure is “not explicitly named COS in the paper,” but is described through that lens because it “suppresses empirical-process oscillations” and “exploits convexity” via essential subsets, star-hulls, and convex averaging (Mendelson, 2017). The triangular-mesh DG paper similarly “does not use the term ‘COS’,” and reconstructs a COS-like procedure by combining oscillation-eliminating methodology with optimal convex decomposition (Ding et al., 2024). By contrast, the EPO framework explicitly defines an “oscillation-suppressing (COS) constraint” as one of three convex constraints enforced on a cellwise candidate state (Wu, 31 Mar 2026). The computational-probability literature uses COS in a different sense altogether: a Fourier–cosine expansion method for densities and transforms (Wang et al., 2 Dec 2025).

2. Statistical learning: unrestricted learning through convex localization

For learning under squared loss, the COS interpretation in "An optimal unrestricted learning procedure" considers an arbitrary class $F$ of square-integrable functions on $(\Omega,\mu)$, a distribution $X \sim \mu$, and an unknown square-integrable target $Y$. The loss is

$L(f(X),Y)=(f(X)-Y)^2,$

with risk

$R(f)=\mathbb{E}(f(X)-Y)^2,$

and oracle

$f^*=\arg\min_{f\in F}R(f).$

The analysis introduces $\sigma^2=\mathbb{E}(f^*(X)-Y)^2$, a uniform integrability assumption on $U-U$ with $U=(F+F)/2$, and the target-class interaction parameter

$(\Omega,\mu)$0

which ensures

$(\Omega,\mu)$1

Localization is carried out through the star-shaped set

$(\Omega,\mu)$2

which enables convex localization even when $(\Omega,\mu)$3 is non-convex (Mendelson, 2017).

The procedure $(\Omega,\mu)$4 has two components, $(\Omega,\mu)$5 and $(\Omega,\mu)$6, and is run twice. $(\Omega,\mu)$7 is a distance oracle: for $(\Omega,\mu)$8, it computes $(\Omega,\mu)$9, the $X \sim \mu$0-th largest entry of the nonincreasing rearrangement of $X \sim \mu$1. $X \sim \mu$2 is a tournament on a second independent sample partitioned into $X \sim \mu$3 blocks of size $X \sim \mu$4. For each block $X \sim \mu$5,

$X \sim \mu$6

and

$X \sim \mu$7

The “home-and-away” decision rule declares $X \sim \mu$8 if, on more than $X \sim \mu$9 blocks, either $Y$0 when $Y$1, or $Y$2 when $Y$3. The winners form

$Y$4

The algorithm first applies $Y$5 on $Y$6 to obtain $Y$7, then forms $Y$8, repeats $Y$9 on $L(f(X),Y)=(f(X)-Y)^2,$0, obtains $L(f(X),Y)=(f(X)-Y)^2,$1, and outputs any $L(f(X),Y)=(f(X)-Y)^2,$2 (Mendelson, 2017).

Its oscillation-suppressing content is encoded by localized intrinsic and extrinsic complexity measures,

$L(f(X),Y)=(f(X)-Y)^2,$3

defined through localized Rademacher and multiplier averages over $L(f(X),Y)=(f(X)-Y)^2,$4. The fixed-point prescription is to choose $L(f(X),Y)=(f(X)-Y)^2,$5 so that these localized oscillations are at most proportional to $L(f(X),Y)=(f(X)-Y)^2,$6 and $L(f(X),Y)=(f(X)-Y)^2,$7, respectively. On high-probability events, $L(f(X),Y)=(f(X)-Y)^2,$8 becomes an $L(f(X),Y)=(f(X)-Y)^2,$9-essential subset with $R(f)=\mathbb{E}(f(X)-Y)^2,$0 and $R(f)=\mathbb{E}(f(X)-Y)^2,$1, and the two-stage convexification yields

$R(f)=\mathbb{E}(f(X)-Y)^2,$2

The main theorem states that if

$R(f)=\mathbb{E}(f(X)-Y)^2,$3

with $R(f)=\mathbb{E}(f(X)-Y)^2,$4 defined from $R(f)=\mathbb{E}(f(X)-Y)^2,$5, $R(f)=\mathbb{E}(f(X)-Y)^2,$6, and the term

$R(f)=\mathbb{E}(f(X)-Y)^2,$7

then with probability at least $R(f)=\mathbb{E}(f(X)-Y)^2,$8,

$R(f)=\mathbb{E}(f(X)-Y)^2,$9

The sample complexity matches, up to constants, the lower bounds required to overcome intrinsic and extrinsic obstructions. If $f^*=\arg\min_{f\in F}R(f).$0 is convex, then $f^*=\arg\min_{f\in F}R(f).$1, so the procedure is proper; if $f^*=\arg\min_{f\in F}R(f).$2 is non-convex, the output lies in a convex superset and the procedure is unrestricted. The heavy-tail robustness comes from order-statistic distance estimation in $f^*=\arg\min_{f\in F}R(f).$3 and majority-of-block control in $f^*=\arg\min_{f\in F}R(f).$4 (Mendelson, 2017).

3. Finite volume and discontinuous Galerkin stabilization: the EPO ray and COS radius

In "EPO: A Unified Framework for Entropy Stability, Positivity, and Oscillation Suppression," COS is an explicit local constraint acting on a candidate finite volume or discontinuous Galerkin update for a hyperbolic system $f^*=\arg\min_{f\in F}R(f).$5. On each cell $f^*=\arg\min_{f\in F}R(f).$6, the candidate nodal array $f^*=\arg\min_{f\in F}R(f).$7 has conservative average

$f^*=\arg\min_{f\in F}R(f).$8

and all corrections are restricted to the mean-preserving ray

$f^*=\arg\min_{f\in F}R(f).$9

At quadrature nodes,

$\sigma^2=\mathbb{E}(f^*(X)-Y)^2$0

The final limited state is

$\sigma^2=\mathbb{E}(f^*(X)-Y)^2$1

where $\sigma^2=\mathbb{E}(f^*(X)-Y)^2$2 is determined by convex constraints for admissible state, entropy, and oscillation suppression (Wu, 31 Mar 2026).

The admissible-state radius $\sigma^2=\mathbb{E}(f^*(X)-Y)^2$3 is the largest $\sigma^2=\mathbb{E}(f^*(X)-Y)^2$4 such that every node belongs to a closed convex invariant set $\sigma^2=\mathbb{E}(f^*(X)-Y)^2$5. For the compressible Euler equations, a common choice is

$\sigma^2=\mathbb{E}(f^*(X)-Y)^2$6

The entropy radius $\sigma^2=\mathbb{E}(f^*(X)-Y)^2$7 is defined by a convex nodal inequality

$\sigma^2=\mathbb{E}(f^*(X)-Y)^2$8

where $\sigma^2=\mathbb{E}(f^*(X)-Y)^2$9 is a local budget coming from weak cell-average entropy stability. Along the ray,

$U-U$0

is convex and nondecreasing, so

$U-U$1

For quadratic entropy $U-U$2, the paper gives a closed form for $U-U$3. The construction also extends to any prescribed finite family of convex entropy pairs by computing one radius per pair and taking the minimum (Wu, 31 Mar 2026).

The COS constraint itself is posed as a bound-preserving convex local bound at quadrature or nodal points. For each component $U-U$4 and node $U-U$5,

$U-U$6

with practical bounds derived from neighbor averages, a monotone low-order update, or two-point Riemann/Lax–Friedrichs averages. Writing

$U-U$7

the componentwise restriction yields closed-form upper bounds $U-U$8, and the COS radius is

$U-U$9

The final limiter is then

$U=(F+F)/2$0

or, in positivity-first form,

$U=(F+F)/2$1

Because the admissible-state set, entropy sublevel set, and COS set are closed convex sets along the ray, the limiter is the maximal radial projection into their intersection. The framework proves preservation of cell averages, invariant-set preservation, local and global strong entropy inequalities, stagewise budgets for SSP Runge–Kutta methods, an SSP multistep variant, and extensions on rectangular and unstructured triangular meshes (Wu, 31 Mar 2026).

4. Unstructured triangular DG: RIOE damping and optimal convex decomposition

On unstructured triangular meshes, the COS-like construction in "Robust DG Schemes on Unstructured Triangular Meshes: Oscillation Elimination and Bound Preservation via Optimal Convex Decomposition" combines oscillation-eliminating damping with convex bound preservation. The scalar or system conservation law is discretized in the DG space

$U=(F+F)/2$2

with cell average

$U=(F+F)/2$3

The semi-discrete method uses a Lax–Friedrichs flux

$U=(F+F)/2$4

and SSPRK time stepping (Ding et al., 2024).

The oscillation-eliminating operator $U=(F+F)/2$5 is defined by a pseudo-time damping equation whose exact modal solution preserves the average and exponentially damps higher modes: $U=(F+F)/2$6 for $U=(F+F)/2$7. The coefficients

$U=(F+F)/2$8

depend only on the cell and its edge neighbors, and the paper establishes conservation, compactness, scale invariance, and evolution invariance. For systems such as Euler, direct component-wise OE is not rotation-invariant. The rotation-invariant OE (RIOE) procedure therefore redefines momentum damping via normal and tangential edge components,

$U=(F+F)/2$9

and replaces momentum entries by

$(\Omega,\mu)$00

With this change, $(\Omega,\mu)$01 is rotation-invariant whenever the PDE and numerical flux are rotation-invariant (Ding et al., 2024).

Bound preservation is obtained from feasible convex decompositions of cell averages into edge integrals and internal nodes. For $(\Omega,\mu)$02 and $(\Omega,\mu)$03 triangles, the paper constructs optimal decompositions and the associated maximal BP CFL numbers.

Degree	Optimal BP CFL number	Reported gain over classic decomposition
$(\Omega,\mu)$04	$(\Omega,\mu)$05	$(\Omega,\mu)$06–$(\Omega,\mu)$07
$(\Omega,\mu)$08	$(\Omega,\mu)$09	$(\Omega,\mu)$10–$(\Omega,\mu)$11

For $(\Omega,\mu)$12, the optimal feasible decomposition uses

$(\Omega,\mu)$13

with an explicit internal node $(\Omega,\mu)$14. For $(\Omega,\mu)$15, it uses

$(\Omega,\mu)$16

and two internal nodes with equal weights

$(\Omega,\mu)$17

A simplified two-step limiter then rescales the polynomial about the cell average, first to enforce density positivity and then to enforce positivity of specific internal energy (Ding et al., 2024).

The resulting stagewise scheme is

$(\Omega,\mu)$18

All coefficients use only edge-neighboring cell information, so compactness and parallel efficiency are retained. Numerical evidence includes optimal $(\Omega,\mu)$19-th order accuracy for smooth scalar tests, non-oscillatory Burgers solutions, rotation-invariant Euler implosion results, and substantial efficiency gains in forward-facing step, double Mach reflection, and shock diffraction. Average time-step ratios satisfy $(\Omega,\mu)$20 for $(\Omega,\mu)$21 and $(\Omega,\mu)$22 for $(\Omega,\mu)$23, with corresponding CPU-time reductions such as $(\Omega,\mu)$24 h versus $(\Omega,\mu)$25 h for $(\Omega,\mu)$26 and $(\Omega,\mu)$27 h versus $(\Omega,\mu)$28 h for $(\Omega,\mu)$29 in the forward-facing-step case (Ding et al., 2024).

5. Non-convex optimization: inner-convex SCP as an oscillation-suppressing convexification

In "A recursively feasible and convergent Sequential Convex Programming procedure to solve non-convex problems with linear equality constraints," the COS interpretation refers to a sequential convex programming procedure for

$(\Omega,\mu)$30

where $(\Omega,\mu)$31 and $(\Omega,\mu)$32 are non-convex and continuously differentiable, while the equalities are affine. At iteration $(\Omega,\mu)$33, the method builds convex majorizers

$(\Omega,\mu)$34

satisfying

$(\Omega,\mu)$35

and solves the convex subproblem

$(\Omega,\mu)$36

These are inner-convex approximations: feasibility for the convexified constraints implies feasibility for the original problem, and minimizing the surrogate guarantees monotone descent of the true objective (Virgili-Llop et al., 2018).

For difference-of-convex functions $(\Omega,\mu)$37, the surrogate reduces to linearization of the concave part,

$(\Omega,\mu)$38

More generally, the paper develops Taylor-based inner-convex approximations. If

$(\Omega,\mu)$39

the quadratic term keeps only the positive semidefinite part,

$(\Omega,\mu)$40

and higher-order Taylor terms are convexly overestimated through diagonal terms and absolute coefficient sums. The resulting surrogate is

$(\Omega,\mu)$41

If the Taylor series is truncated at order $(\Omega,\mu)$42, the method adds

$(\Omega,\mu)$43

to recover a global upper bound; if majorization fails at the tentative solution, $(\Omega,\mu)$44 is increased and the convex subproblem is re-solved (Virgili-Llop et al., 2018).

The guarantees are recursive feasibility, monotone descent, and convergence to a KKT point, provided the first convex subproblem is feasible. Under local strong convexity of the Lagrangian Hessian at the limit point and local Lipschitz continuity of the Hessian, the procedure has a quadratic local rate: $(\Omega,\mu)$45 The “oscillation-suppressing” role of the convexification is therefore not nodal or modal but iterative: the method eliminates cost increases and constraint-violation oscillations that are typical of pure successive linearization. The paper also reports real-time performance in aerial-vehicle trajectory optimization. With $(\Omega,\mu)$46 nodes and non-convex thrust and keep-out-zone constraints, the implementation was approximately $(\Omega,\mu)$47 faster than GPOPS-II on a laptop; a penalty phase found an admissible solution in approximately $(\Omega,\mu)$48 of cases, and once feasibility was obtained the SCP achieved $(\Omega,\mu)$49 convergence (Virgili-Llop et al., 2018).

6. Fourier–cosine COS method: tail-energy admissibility and convergence

In computational probability and option pricing, COS refers to the Fourier–cosine expansion method. Here the target density $(\Omega,\mu)$50 is approximated on a truncation interval $(\Omega,\mu)$51 by

$(\Omega,\mu)$52

with coefficients

$(\Omega,\mu)$53

If

$(\Omega,\mu)$54

is the characteristic function, practical coefficients are computed as

$(\Omega,\mu)$55

and the $(\Omega,\mu)$56-term approximant is

$(\Omega,\mu)$57

The 2025 convergence note studies the symmetric case $(\Omega,\mu)$58 and defines the tail cosine energy

$(\Omega,\mu)$59

A density $(\Omega,\mu)$60 is COS-admissible if $(\Omega,\mu)$61 as $(\Omega,\mu)$62 (Wang et al., 2 Dec 2025).

The main sufficient condition is moment-based. In one dimension, if

$(\Omega,\mu)$63

and

$(\Omega,\mu)$64

for some $(\Omega,\mu)$65, then $(\Omega,\mu)$66 is COS-admissible and

$(\Omega,\mu)$67

hence

$(\Omega,\mu)$68

In $(\Omega,\mu)$69 dimensions, if

$(\Omega,\mu)$70

for some $(\Omega,\mu)$71, then the $(\Omega,\mu)$72-dimensional tail energy $(\Omega,\mu)$73 satisfies

$(\Omega,\mu)$74

where

$(\Omega,\mu)$75

The paper’s main convergence theorem is

$(\Omega,\mu)$76

for COS-admissible $(\Omega,\mu)$77 (Wang et al., 2 Dec 2025).

This use of COS differs conceptually from the convex-admissibility frameworks in learning, PDE stabilization, and SCP. The relevant “oscillation suppression” here is the control of truncation-induced oscillatory tail contributions through weighted $(\Omega,\mu)$78 moment bounds and blockwise orthogonality arguments. The note enlarges previous admissibility conditions by weakening the one-dimensional moment requirement from $(\Omega,\mu)$79 to any $(\Omega,\mu)$80, and from the multidimensional requirement to any $(\Omega,\mu)$81. As a result, it includes heavy-tailed distributions such as Student $(\Omega,\mu)$82 with small degrees of freedom (Wang et al., 2 Dec 2025).

7. Cross-cutting principles, distinctions, and common misconceptions

Across these literatures, the recurrent technical pattern is the use of convex structure to regularize unstable behavior. In unrestricted learning, convexity enters through star-hulls, essential subsets, and midpoint averaging, while oscillation suppression is achieved by majority-of-block control of quadratic and multiplier empirical processes (Mendelson, 2017). In EPO, convexity is literal geometry on a mean-preserving ray, and oscillation suppression is a nodal bound-preserving constraint intersected with positivity and entropy sublevel sets (Wu, 31 Mar 2026). In triangular-mesh DG, oscillation elimination is modal damping, while convexity appears in the feasible decomposition of cell averages and in the convex rescaling limiter (Ding et al., 2024). In SCP, convexity arises from inner-convex surrogates, and the suppressed oscillations are iterative cost increases and feasibility losses (Virgili-Llop et al., 2018).

A central misconception is to treat COS as a single canonical procedure. The sources do not support that interpretation. Several papers explicitly state that the relevant method is being presented “through the lens” of COS or reconstructed as “COS-like,” whereas the EPO framework contains an explicit COS constraint, and the Fourier–cosine literature uses COS as an acronym for a different numerical expansion method. This suggests that “COS procedure” is an umbrella label covering at least four distinct mathematical mechanisms: robust aggregation, radial projection into convex admissible sets, convex decomposition with rescaling, and inner-convex majorization; the Fourier–cosine COS method forms a separate acronymic lineage rather than a variant of those convex-geometric procedures (Mendelson, 2017, Wu, 31 Mar 2026, Virgili-Llop et al., 2018, Wang et al., 2 Dec 2025).

A second distinction concerns what is being suppressed. In learning, the target is empirical-process oscillation under heavy tails. In hyperbolic discretization, it is spurious oscillation at nodal or modal level, often tied to positivity and entropy stability. In optimization, it is oscillatory or non-monotone iterate behavior. In Fourier–cosine approximation, it is truncation error encoded by tail cosine energy. The shared terminology is therefore organizational rather than ontological: the underlying objects, guarantees, and proof techniques differ substantially even when convexity and oscillation suppression are both present.