---
title: Two-Step Convex Relaxation and Recovery (TS-CRR)
url: https://www.emergentmind.com/topics/two-step-convex-relaxation-and-recovery-ts-crr
type: topic
---

# Two-Step Convex Relaxation and Recovery (TS-CRR)

Two-Step Convex Relaxation and Recovery (TS-CRR) denotes a methodological pattern in which a nonconvex, combinatorial, or otherwise intractable estimation problem is first replaced by a convex surrogate and then mapped back to the original object by an explicit recovery step. The label appears explicitly in sparse polynomial regression with anomalous data filtering, where a convex relaxation of a fractional program is followed by coefficient recovery on the selected monomial support [2508.18199]. Earlier papers do not generally use the name, but they instantiate closely related architectures: lifted semidefinite relaxation followed by factor extraction in phase retrieval [1109.4499], autocorrelation relaxation followed by KL-based deautocorrelation for pairwise interactions [1511.03354], total-variation relaxation followed by support extraction from a dual polynomial in off-the-grid super-resolution [1711.08239], and staged weighted \(L_1\) refinement for sparse feature selection [1106.0565].

## 1. Conceptual scope and historical placement

In the literature represented here, TS-CRR is not a single fixed algorithm but a family of architectures organized around a common division of labor. The relaxation stage enlarges the feasible set or replaces a discrete structural prior by a convex proxy; the recovery stage then extracts support, amplitudes, factors, or exact consistency constraints from the relaxed solution. This suggests that TS-CRR is best understood as a design principle rather than as a single model class.

The explicit 2025 formulation for sparse polynomial regression is the first paper in this set to use the term “Two-Step Convex Relaxation and Recovery” directly [2508.18199]. Closely related antecedents appear under other names. “PhaseLift” lifts a quadratic inverse problem into a rank-one matrix problem and then extracts the signal by eigendecomposition [1109.4499]. “Relax, Compensate and then Recover” (RCR) is a three-phase graphical-model formalism in which relaxed equivalence constraints are selectively restored to tighten a dual-decomposition approximation [1504.01173]. “Multi-stage convex relaxation” in sparse regression and structured low-rank recovery goes beyond a literal two-step procedure, but its first stage is an ordinary convex relaxation and its later stages act as recovery or debiasing refinements [1106.0565; 1703.03898].

A recurrent distinction across these papers is whether recovery is explicit, implicit, or internalized. In some cases the recovery stage is operationally separate, as in support localization from a dual trigonometric polynomial in two-dimensional super-resolution [1711.08239] or KL-based deautocorrelation in pairwise interaction energies [1511.03354]. In others, the convex optimizer already equals the target structured object, so the second step reduces to reading off a support or a rank-one factor, as in planted dense subgraph recovery [1305.4891]. A further variant appears in convex relaxations of shallow convolutional networks, where a randomized perturbation selects the desired optimizer inside the convex program itself rather than through a separate post-processing stage [1901.00035].

## 2. Explicit TS-CRR in sparse polynomial regression under anomalous data

The most explicit TS-CRR formulation in this corpus is the sparse polynomial regression model with anomalous data filtering [2508.18199]. The data are \(\{x^{(k)}\}_{k=1}^N\) and \(\{y^{(k)}\}_{k=1}^N\), with \(x^{(k)}\in\mathbb R^n\), and the regression model is
\[
y^{(k)}=\sum_{\alpha \in \Gamma_d} c_\alpha \, \big(x^{(k)}\big)^\alpha + \epsilon^{(k)},\qquad k=1,\dots,N,
\]
where
\[
\Gamma_d=\left\{\alpha\in \mathbb{N}^n \mid \sum_{i=1}^n \alpha_i \le d\right\}.
\]
The baseline fitting criterion is minimax:
\[
\min_{c,\gamma}\ \gamma
\quad\text{s.t.}\quad
\left|y^{(k)}-\sum_{\alpha\in\Gamma_d} c_\alpha (x^{(k)})^\alpha\right|\le \gamma,\ \forall k.
\]

Sparsity and anomaly filtering are encoded jointly through binary monomial-selection variables \(s_\alpha\in\{0,1\}\) and sample-inclusion variables \(b_k\in\{0,1\}\). The resulting model imposes
\[
-M(1-b_k)+\left|y^{(k)}-\sum_{\alpha\in\Gamma_d} c_\alpha (x^{(k)})^\alpha\right|\le \gamma,
\]
\[
-M s_\alpha \le c_\alpha \le M s_\alpha,
\]
together with the budgets
\[
\sum_{k=1}^N b_k = l_b,\qquad \sum_{\alpha\in\Gamma_d} s_\alpha = l_m.
\]
Thus \(N-l_b\) is the anomaly budget and \(l_m\) is the monomial budget.

The paper then passes through a continuous QCQP reformulation and an equivalent fractional program (FP). Its central structural result is that the FP has a convex objective and convex constraints except for the single scalar nonconvex constraint
\[
\|\hat s\|^2+\|\hat b\|^2+\hat v^2\ge 1,
\]
provided
\[
\rho \ge 1+\frac{\sqrt{m_d+N}}{2}.
\]
The TS-CRR algorithm relaxes that remaining nonconvexity by the linear inequality
\[
\sum_{\alpha\in\Gamma_d}\hat s_\alpha+\sum_{k=1}^N \hat b_k+\hat v\ge 1.
\]
The paper proves that this linear relaxation is valid, tighter than the SDC-based relaxation, and therefore also tighter than the SOC-based relaxation [2508.18199].

The two stages are then explicit. In step 1, the linear-based convex relaxation of the FP is solved, yielding \((\hat c,\hat s,\hat b,\hat\gamma,\hat v)\). These are mapped back through \(\mathcal T^{-1}\) to produce \((s^*,b^*,\gamma^*)\), which the paper describes as identifying the sparse monomial structure and anomaly-filtering pattern. In step 2, the coefficients are recovered by solving the LP
\[
\min_{c,\gamma}\ \gamma
\quad\text{s.t.}\quad
\left| y^{(k)}-\sum_{\alpha\in\Gamma_d} s_\alpha^* c_\alpha (x^{(k)})^\alpha \right| \le \gamma,\qquad \forall k.
\]
An exactness condition is also stated: if
\[
\|\hat{s}^*\|^2 + \|\hat{b}^*\|^2 + (\hat{v}^*)^2
=
\sum_{\alpha \in \Gamma_d} \hat{s}_\alpha^* + \sum_{k=1}^N \hat{b}_k^* + \hat{v}^*,
\]
then the linear relaxation solves the original FP exactly.

The empirical evaluation uses two datasets. For Nordic electricity prices, the reported TS-CRR parameters are
\[
d=2,\qquad N=360,\qquad m_d=1485,\qquad N-l_b=4,\qquad l_m=120,
\]
and the paper reports mean interpolation MSE \(0.0001\), mean extrapolation MSE \(0.0003\), and max extrapolation MSE \(0.0069\), compared with linear-regression values \(0.0002\), \(0.0002\), \(0.0089\) and polynomial-regression values \(0.0001\), \(0.0005\), \(0.0138\) [2508.18199]. For temperature forecasting, the reported parameters are
\[
d=4,\qquad N=360,\qquad m_d=35,\qquad N-l_b=6,\qquad l_m=32.
\]
The paper describes the method as more balanced in interpolation and extrapolation than several standard regression and AI baselines.

## 3. Convex-relaxation architectures associated with TS-CRR

The convex stage in TS-CRR varies substantially across domains, but the recurring pattern is replacement of a hard structural prior by a tractable convex surrogate.

In phase retrieval, lifting replaces a quadratic inverse problem in \(x\) by a linear inverse problem in
\[
X=xx^*,
\]
with measurements
\[
\mathcal A(X)_i = z_i^*Xz_i = \operatorname{Tr}(z_i z_i^*X).
\]
The corresponding relaxation is the semidefinite program
\[
\min_{X\succeq 0}\operatorname{Tr}(X)
\quad\text{s.t.}\quad
\mathcal A(X)=b,
\]
or its noisy version with \(\|\mathcal A(X)-b\|_2\le \epsilon\) [1109.4499]. In sparse quadratic recovery, the lifted variable is again \(X=xx^T\), but sparsity is added through the mixed objective
\[
\min_X \ \|X\|_1 + \lambda \operatorname{Tr}(X)
\quad\text{s.t.}\quad
z_j^T X z_j = b_j,\ \forall j,\quad X\succeq 0,
\]
which the paper interprets as a sparse variant of PhaseLift [1209.4785].

In off-the-grid super-resolution, the convex stage is infinite-dimensional but still classical: total-variation minimization over measures,
\[
\min_{z_{2D}}\|z_{2D}\|_{\mathrm{TV}}
\quad\text{s.t.}\quad
\mathcal F_{2D}z_{2D}=\mathbf Y,
\]
with a dual formulation in terms of a bounded low-pass trigonometric polynomial [1711.08239]. In planted dense subgraph recovery, the nonconvex rank-and-cardinality problem is relaxed to
\[
\min \left\{ \|X\|_*+ \gamma \|Y\|_1 : \; e^T X e = k^2,\; X_{ij} + Y_{ij} = 0  \ \forall ij \in \tilde E,\; X \in [0,1]^{V\times V} \right\},
\]
so the convex stage is a low-rank-plus-sparse program in the style of robust PCA [1305.4891].

Other TS-CRR-related relaxations operate on still different surrogate spaces. Pairwise interaction minimization is reformulated over autocorrelations \(F=\rho\circ\rho\), and the exact but nonconvex autocorrelation set is enlarged to a convex cone-slice \(\mathcal C\), leading to the linear conic program
\[
(R)\quad \min_{F\in\mathcal C}\frac12\langle F,W\rangle
\]
with positivity, Fourier-sign, symmetry, and mass constraints [1511.03354]. In shallow ReLU networks, the relaxation is an epigraph relaxation in the original parameter space,
\[
\min_{w,z}\sum_{i=1}^n (z_i-y_i)^2
\quad\text{s.t.}\quad
z_i\ge \sigma(x_i^T w),
\]
augmented by a random perturbation \(\beta r^T w\) because the naive relaxation has spurious optima [1901.00035].

## 4. Recovery mechanisms

The recovery stage in TS-CRR is domain-dependent because the relaxed variable usually lives in a different space from the original object. In PhaseLift, the SDP returns a PSD matrix \(\hat X\), not a signal vector. Recovery therefore consists of rank-one factor extraction:
\[
\hat X = \sum_{k=1}^n \hat\lambda_k \hat u_k \hat u_k^*,
\qquad
\hat x = \sqrt{\hat\lambda_1}\,\hat u_1.
\]
In the exact case \(\hat X=xx^*\), this yields \(x\) up to global phase; in noise, the top eigenpair provides the approximate signal [1109.4499].

In two-dimensional super-resolution, recovery is mediated by the dual polynomial
\[
Q(\mathbf t)=\sum_{\mathbf k\in J} q_{\mathbf k} e^{j2\pi\langle \mathbf t,\mathbf k\rangle}.
\]
The support is localized from the rule
\[
\hat T = \left\{\mathbf t\in [0,1]^2:\left|(\mathcal F_{2D}^*\hat{\mathbf C})(\mathbf t)\right|=1\right\},
\]
and amplitudes are then obtained from the Fourier fitting system
\[
y_{\mathbf k} = \sum_{\mathbf t_i\in \hat T} d_i e^{-j2\pi\langle \mathbf k,\mathbf t_i\rangle}.
\]
The paper emphasizes that the certificate proving optimality is also the practical support-localization device [1711.08239].

In pairwise interaction problems, the relaxed variable is the autocorrelation \(F_R\), and recovery seeks a realizable probability measure \(\rho\) whose autocorrelation matches \(F_R\). The criterion is relative entropy,
\[
\mathcal F(\rho)= \int_\Omega F_R(\mathbf x)\log\frac{F_R(\mathbf x)}{(\rho\circ\rho)(\mathbf x)}\,d\mathbf x,
\]
and the associated Schulz–Snyder iteration is
\[
\rho_{n+1}(\mathbf{x}) = \rho_n(\mathbf{x})\int_\Omega \rho_n(\mathbf{x}+\mathbf{y})\frac{F_R(\mathbf y)}{F_{\rho_n}(\mathbf y)}\,d\mathbf y.
\]
Here recovery is not a support readout but a deautocorrelation problem on the manifold of realizable autocorrelations [1511.03354].

Sparse polynomial regression provides a more classical support-first recovery pattern. After the linear-based convex relaxation identifies \(s^*\), the recovery stage solves an LP on the selected monomial support, rather than attempting to estimate coefficients directly inside the relaxed model [2508.18199]. In graphical-model inference, RCR interprets recovery as restoration of previously relaxed equivalence constraints \(eq(X,X_i)\), which tightens the dual upper bound and can lead all the way back to exact inference [1504.01173].

A useful contrast appears in planted dense subgraph recovery. There, under the theorem assumptions, the convex optimizer itself is already
\[
X^*=vv^T,\qquad Y^*=-P_{\tilde E}(vv^T),
\]
so the “recovery” step is essentially trivial: one reads off the planted vertex set from the support or rank-one factor of \(X^*\) [1305.4891]. This suggests that TS-CRR spans both explicit two-stage pipelines and one-shot convex programs whose solutions already encode the recovered object.

## 5. Exactness, sample complexity, and bias reduction

A central reason for the TS-CRR pattern is that the convex stage often has sharp certificates, while the recovery stage either preserves or improves them. In phase retrieval, if
\[
m \ge c_0\, n\log n,
\]
then the trace-minimization SDP has the unique solution \(xx^*\) with probability at least
\[
1 - 3 e^{-\gamma m/n},
\]
and in the noisy case the lifted estimator obeys
\[
\|\hat X - xx^*\|_2 \le C_0 \epsilon
\]
with corresponding vector-level stability after top-eigenvector extraction [1109.4499].

In sparse quadratic recovery, exact recovery by the mixed lifted convex program is proved when
\[
m \ge O(k^2 \log n),
\]
equivalently
\[
k \le O\!\left(\sqrt{\frac{m}{\log n}}\right),
\]
but the same paper also proves that for this class of naive convex relaxations one cannot expect exactness unless
\[
m \ge O\!\left(\frac{k^2}{\log^2 n}\right)
\]
for flat \(k\)-sparse signals [1209.4785]. This negative result is especially important for TS-CRR because it motivates separating support identification from final signal recovery instead of enforcing all structure in a single lifted convex model.

In off-the-grid super-resolution, the exact TV-minimization theorem is certificate-based: if \(f_c \ge 2\times 10^3\) and
\[
\Delta(T)\ge 1.68\,\lambda_c,\qquad \lambda_c=\frac{1}{f_c},
\]
then the TV solution is unique [1711.08239]. The support-extraction stage is justified by the same dual polynomial that certifies exactness.

The sparse feature-selection paper makes the bias issue explicit. Standard Lasso has \(\ell_2\)-error
\[
\|\hat{\mathbf w}_{L_1}-\bar{\mathbf w}\|_2 = O(\sqrt{\bar{k}}\,\lambda),
\]
and thresholding therefore demands the stronger beta-min condition
\[
\min_{j\in\bar F} |\bar w_j| \gtrsim \sigma\sqrt{\frac{\bar{k}\ln p}{n}}.
\]
By contrast, the multi-stage weighted \(L_1\) scheme based on capped-\(L_1\) regularization removes penalty from coefficients above a threshold and achieves exact support recovery after \(O(\log \bar{k})\) stages under sub-Gaussian noise and sparse-eigenvalue assumptions [1106.0565]. The paper explicitly states that a literal two-stage method does not remove the bias issue in general under RIP.

An analogous debiasing phenomenon appears in noisy structured low-rank matrix recovery. The first stage is exactly nuclear-norm minimization, while later stages solve
\[
X^k\in\arg\min_X\left\{\|X\|_*-\langle W^{k-1},X\rangle:\ \|\mathcal A(X)-b\|\le\delta,\ X\in\Omega\right\},
\]
where \(W^{k-1}\) is built from previous singular structure. Under a restricted eigenvalue condition, the paper shows deterministic reduction of both estimation error and approximate-rank bounds in later stages, together with geometric convergence to a statistical floor [1703.03898]. In TS-CRR language, stage 2 is a convex recovery correction that reduces nuclear-norm bias.

## 6. Limitations, misconceptions, and broader interpretation

A common misconception is that TS-CRR always means exactly two convex programs. The literature here is more heterogeneous. Some methods are strictly two-stage, such as sparse polynomial regression with linear-based relaxation followed by an LP [2508.18199]. Others are multi-stage by design: sparse feature selection proves that more than two stages are generally needed for exact unbiased recovery under RIP, although only \(O(\log \bar{k})\) stages suffice [1106.0565], and structured low-rank recovery derives an entire sequence of convex corrections from an exact-penalty reformulation [1703.03898].

Another misconception is that the recovery stage is always a rounding heuristic. In several papers it is theorem-level mathematics. The dual polynomial in super-resolution is simultaneously an optimality certificate and a support-localization mechanism [1711.08239]. The KL recovery stage in pairwise interactions comes with the sufficient condition \(\mathcal F(\rho^*)=0\), which implies exact global optimality, and lattice Dirac solutions are recovered exactly when \(F_R\circ F_R=F_R\) [1511.03354]. Conversely, some papers show that recovery may be only implicit or even absent as a separate algorithmic object: planted dense subgraph recovery returns the planted rank-one matrix directly [1305.4891], while shallow CNN relaxation performs recovery through randomized tie-breaking inside the convex program rather than through a separate decoding stage [1901.00035].

Computational cost is a persistent limitation. Lifted semidefinite models operate in \(O(n^2)\)-scale matrix spaces in phase retrieval [1109.4499] and sparse quadratic recovery [1209.4785]. Two-dimensional super-resolution solves a dual SDP whose \(\mathbf Q_0\) variable is of size \(n^2\times n^2\), and the paper explicitly notes poor scaling at large bandwidths [1711.08239]. The 2025 TS-CRR paper improves tractability by isolating the FP’s nonconvexity to a single scalar quadratic constraint, but it still requires choosing \(M\), \(l_m\), \(l_b\), and a sufficiently large \(\rho\), and it does not provide a sophisticated rounding rule beyond the inverse mapping \(\mathcal T^{-1}\) [2508.18199].

The broader lesson is that TS-CRR is most effective when the relaxation stage produces a surrogate object with clear structural semantics: a rank-one PSD matrix, an autocorrelation, a dual polynomial, a weighted sparse estimate, or a partially decomposed graphical model. The recovery stage then exploits that semantics rather than treating the convex solution as an endpoint. This suggests that the central design question in TS-CRR is not merely how to convexify, but how to choose a convex surrogate from which the target structure remains recoverable.

Source: https://www.emergentmind.com/topics/two-step-convex-relaxation-and-recovery-ts-crr