Gaussian Randomized Rounding

• Gaussian randomized rounding is a family of techniques that uses Gaussian noise to convert continuous relaxations into discrete representations while preserving key expectations and correlations.
• It leverages methods such as multidimensional Brownian motion, hyperplane rounding, and Grothendieck-identity-based techniques to achieve unbiased outputs with controlled variance.
• Applications span combinatorial optimization, semidefinite programming, covariance approximation, computer vision, quantum computing, and differential privacy.

Searching arXiv for the cited paper and closely related randomized-rounding work. Gaussian randomized rounding denotes a family of randomized discretization procedures in which Gaussian structure is used to convert continuous or relaxed solutions into discrete outputs while retaining analytically controlled expectations, correlations, or objective values. In the literature represented here, the term spans several distinct but related constructions: multidimensional Brownian-motion rounding for packing integer programs, Gaussian hyperplane rounding in the Goemans–Williamson framework, Grothendieck-identity-based sign rounding for covariance and positive-semidefinite regularity, and Gaussian-compatible probabilistic quantization in heatmap regression. Across these settings, the unifying feature is that Gaussian randomness is not incidental noise but the central mechanism by which a continuous object—an LP or SDP solution, a covariance structure, a kernel, or a sub-pixel coordinate—is mapped to a discrete representation with provable preservation properties (Sen, 2014, Boedihardjo et al., 2023).

1. Conceptual scope and core definitions

Gaussian randomized rounding is not a single algorithm but a class of rounding paradigms. In packing integer programming, the procedure is formulated as a multidimensional Brownian motion in $\mathbb{R}^n$ started at an optimal fractional feasible solution $\bar{x}$ and absorbed at the vertices of the hypercube, producing a distribution over $P \subset \{0,1\}^n$ whose expected value for any linear objective matches the value at $\bar{x}$ (Sen, 2014). In semidefinite-programming settings such as MAX-CUT, one samples a Gaussian vector and rounds by signs of projections, yielding the classical Gaussian or hyperplane randomized rounding associated with Goemans–Williamson (Shi et al., 2024). In covariance-loss analysis for positive semidefinite matrices and kernels, one samples independent Gaussian directions, records sign patterns, and applies a sine correction derived from Grothendieck’s identity to preserve inner products in expectation (Boedihardjo et al., 2023).

The shared mathematical principle is expectation or correlation preservation under a Gaussian mechanism. In the Brownian-motion formulation, each coordinate evolves as a martingale until absorption, so linear objectives are preserved in expectation (Sen, 2014). In Gaussian hyperplane rounding, pairwise sign correlations are linked to continuous correlations through the arcsine law, enabling approximation-ratio analysis for two-body objectives (Shi et al., 2024). In the covariance-loss setting, Gaussian sign features induce partitions with at most $2^r$ cells, and the estimator

$$F_g(x, x') := \sin\!\left(\frac{\pi}{2}\, S(x, x')\right)$$

is unbiased for $\langle x,x'\rangle$ under the stated assumptions (Boedihardjo et al., 2023).

This breadth of usage explains why “Gaussian randomized rounding” can refer to markedly different operational procedures. A plausible implication is that the phrase is best understood by the invariants being preserved—linear expectation, pairwise correlation, covariance structure, or continuous coordinates—rather than by any single sampling rule.

2. Brownian-motion rounding for packing integer programs

For packing integer programs with nonnegative data, the continuous formulation in the source material begins from the LP relaxation

$$\max c^\top x \quad \text{subject to } A x \le b,\; x\in[0,1]^n,$$

with an optimal solution $\bar{x}$ satisfying $A\bar{x}\le b$ (Sen, 2014). The Gaussian or Brownian rounding process defines a continuous-time stochastic trajectory in $\bar{x}$0,

$\bar{x}$1

with absorption at $\bar{x}$2 and $\bar{x}$3 coordinatewise; once a coordinate hits the boundary it is frozen there (Sen, 2014). The stopping time

$\bar{x}$4

marks arrival at a vertex of the hypercube (Sen, 2014).

The analytical justification is martingale-based. For any linear functional $\bar{x}$5, the process is a martingale up to $\bar{x}$6, and optional stopping yields

$\bar{x}$7

which gives unbiased objective preservation (Sen, 2014). For each packing constraint row $\bar{x}$8, the process $\bar{x}$9 is likewise a martingale, so the expectation of each constraint remains bounded by its LP value (Sen, 2014). Constraint control then proceeds through predictable quadratic variation and Gaussian or martingale tail bounds. The source material records the quadratic variation form

$P \subset \{0,1\}^n$0

and the resulting tail estimate

$P \subset \{0,1\}^n$1

where $P \subset \{0,1\}^n$2 is the total quadratic variation term used in the analysis (Sen, 2014).

The 2015 development makes the algorithmic intent more explicit. “Randomised Rounding with Applications” describes iterative randomized rounding for packing constraints $P \subset \{0,1\}^n$3 with $P \subset \{0,1\}^n$4, using a multidimensional Brownian walk that gradually fixes coordinates near $P \subset \{0,1\}^n$5 or $P \subset \{0,1\}^n$6 and then finishes with the constructive Lovász Local Lemma (Madan et al., 2015). The Brownian stage preserves marginals, while sparsifying the active constraint system so that each remaining row has at most $P \subset \{0,1\}^n$7 unfixed variables before the final LLL phase (Madan et al., 2015). The paper states that independent randomized rounding violates constraints by at most $P \subset \{0,1\}^n$8, whereas the iterative Brownian method can exploit the reduced dependencies of the sparser system through the Lovász Local Lemma and yields improved guarantees in random-row models (Madan et al., 2015).

This line of work positions Gaussian randomized rounding as an alternative to the classical Raghavan–Thompson scheme. The salient distinction is dependence structure: instead of rounding variables independently, the Brownian walk introduces correlated evolution, and that correlation can be shaped through covariance choices and discrepancy-style projections (Sen, 2014, Madan et al., 2015).

3. Gaussian hyperplane rounding and semidefinite relaxations

A second major meaning of Gaussian randomized rounding is the hyperplane-sign construction that arises from semidefinite relaxations. In the MAX-CUT formulation, one solves an SDP with unit vectors $P \subset \{0,1\}^n$9 or, equivalently, a positive semidefinite matrix $\bar{x}$0 with diagonal entries equal to $\bar{x}$1, then samples either a random vector uniformly from the sphere or a Gaussian $\bar{x}$2 and sets

$\bar{x}$3

(Shi et al., 2024). Rotational invariance makes the Gaussian and spherical samplers equivalent for sign patterns (Shi et al., 2024).

The central geometric identity is that if $\bar{x}$4, then the probability of separation under a random hyperplane is

$\bar{x}$5

and the Goemans–Williamson inequality yields the approximation factor

$\bar{x}$6

(Shi et al., 2024). The 2024 robust MAX-CUT work emphasizes that the same randomized rounding framework extends to robust and distributionally robust settings because the edgewise guarantee depends only on correlations and not on the weights themselves (Shi et al., 2024). This preserves the $\bar{x}$7 approximation bound for robust and distributionally robust counterparts of MAX-CUT, and analogous extensions are described for Max-DiCut, MAX-SAT, and Max-2SAT under the same randomization-projection framework (Shi et al., 2024).

The 2025 quantum-circuit application adapts the same Gaussian-sign idea to noisy optimization circuits. There, one estimates two-qubit marginals $\bar{x}$8, projects them to a valid correlation matrix if needed, samples $\bar{x}$9, and rounds by signs $2^r$0 (Martinez et al., 29 Jul 2025). The key identity recorded in that source is again

$2^r$1

for jointly Gaussian coordinates with correlation $2^r$2 (Martinez et al., 29 Jul 2025). For Max-Cut under local depolarizing noise, the paper states that the sampler achieves an approximation ratio $2^r$3, and that on IBMQ hardware the rounded samples “faithfully reproduce the full energy distribution” of the noisy device for the problem at hand (Martinez et al., 29 Jul 2025).

These SDP-based instances show Gaussian randomized rounding in its most classical optimization form: a Gaussian covariance encodes relaxed pairwise structure, and sign extraction turns that structure into discrete $2^r$4 decisions. Unlike the Brownian-walk literature, the primary preserved quantity is not coordinatewise expectation but objective value or correlation after a nonlinear arcsine or arccosine transformation.

4. Grothendieck-identity rounding for covariance loss and PSD regularity

A third formulation is developed in “Covariance loss, Szemeredi regularity, and differential privacy” (Boedihardjo et al., 2023). Here Gaussian randomized rounding is hyperplane rounding driven by signs of Gaussian projections, but the purpose is not combinatorial optimization. Instead, the method constructs a measurable partition of the sample space that preserves pairwise inner products up to controllable error.

For unit vectors $2^r$5 and $2^r$6, the paper uses Grothendieck’s identity in the form

$2^r$7

equivalently,

$2^r$8

(Boedihardjo et al., 2023). Sampling $2^r$9 independent Gaussian vectors $$F_g(x, x') := \sin\!\left(\frac{\pi}{2}\, S(x, x')\right)$$0 produces sign features

$$F_g(x, x') := \sin\!\left(\frac{\pi}{2}\, S(x, x')\right)$$1

which generate a partition into at most $$F_g(x, x') := \sin\!\left(\frac{\pi}{2}\, S(x, x')\right)$$2 cells (Boedihardjo et al., 2023). For pairs $$F_g(x, x') := \sin\!\left(\frac{\pi}{2}\, S(x, x')\right)$$3, the empirical sign average $$F_g(x, x') := \sin\!\left(\frac{\pi}{2}\, S(x, x')\right)$$4 is passed through the sine map to form the “Grothendieck estimator”

$$F_g(x, x') := \sin\!\left(\frac{\pi}{2}\, S(x, x')\right)$$5

which is unbiased for $$F_g(x, x') := \sin\!\left(\frac{\pi}{2}\, S(x, x')\right)$$6 in the unit case (Boedihardjo et al., 2023).

The resulting theorem states that if $$F_g(x, x') := \sin\!\left(\frac{\pi}{2}\, S(x, x')\right)$$7 almost surely, then for any $$F_g(x, x') := \sin\!\left(\frac{\pi}{2}\, S(x, x')\right)$$8 there exists a partition into at most $$F_g(x, x') := \sin\!\left(\frac{\pi}{2}\, S(x, x')\right)$$9 parts such that, for $\langle x,x'\rangle$0,

$\langle x,x'\rangle$1

(Boedihardjo et al., 2023). The corresponding covariance-loss bound is

$\langle x,x'\rangle$2

and the source records that the $\langle x,x'\rangle$3 rate for partitions into at most $\langle x,x'\rangle$4 parts is sharp up to constants (Boedihardjo et al., 2023).

This same rounding mechanism yields weak Szemerédi regularity for PSD matrices and kernels in Hilbert–Schmidt norm. For an $\langle x,x'\rangle$5 PSD matrix $\langle x,x'\rangle$6 with $\langle x,x'\rangle$7, there exists a partition into $\langle x,x'\rangle$8 blocks and a block-constant matrix $\langle x,x'\rangle$9 such that

$$\max c^\top x \quad \text{subject to } A x \le b,\; x\in[0,1]^n,$$0

with $$\max c^\top x \quad \text{subject to } A x \le b,\; x\in[0,1]^n,$$1 obtained by averaging entries over each block (Boedihardjo et al., 2023). The source explicitly contrasts this with classical Frieze–Kannan regularity: the norm is stronger, but positive semidefiniteness is essential, and the Hadamard-matrix example shows that non-PSD matrices do not admit an analogous nontrivial Hilbert–Schmidt bound (Boedihardjo et al., 2023).

A notable feature of this version of Gaussian randomized rounding is that Grothendieck’s constant does not enter the analysis; the paper leverages Grothendieck’s identity directly, and the constants come from the Lipschitz constant of $$\max c^\top x \quad \text{subject to } A x \le b,\; x\in[0,1]^n,$$2 and variance or Hoeffding bounds (Boedihardjo et al., 2023).

5. Structured applications beyond combinatorial optimization

Although the phrase most often appears in optimization and SDP rounding, the supplied literature includes applications in computer vision, privacy, quantum computation, and energy disaggregation.

In heatmap regression, randomized rounding is used as a quantization mechanism for sub-pixel landmark localization rather than as a combinatorial optimizer. “Heatmap Regression via Randomized Rounding” proposes a quantization system induced by randomized rounding that encodes the fractional part of numerical coordinates probabilistically during training and decodes coordinates from a set of activation points during testing (Yu et al., 2020). For a coordinate with fractional parts $$\max c^\top x \quad \text{subject to } A x \le b,\; x\in[0,1]^n,$$3 in heatmap units, the method assigns bilinear probabilities to the four neighboring grid points,

$$\max c^\top x \quad \text{subject to } A x \le b,\; x\in[0,1]^n,$$4

and then either samples a location or constructs a deterministic mixture of four Gaussian heatmaps centered at the integer neighbors (Yu et al., 2020). The paper states that the system is unbiased and lossless in the ideal case, and that the decoder recovers the exact continuous coordinate when the predicted heatmap equals the ground-truth activation probabilities (Yu et al., 2020). Here the “Gaussian” attribute refers to Gaussian-shaped heatmaps rather than Gaussian random vectors; the source explicitly notes that distinction (Yu et al., 2020).

In differential privacy, the covariance-loss paper uses Gaussian randomized rounding to construct partitions with at most $$\max c^\top x \quad \text{subject to } A x \le b,\; x\in[0,1]^n,$$5 cells, which in turn support a synthetic-data mechanism with $$\max c^\top x \quad \text{subject to } A x \le b,\; x\in[0,1]^n,$$6-differential privacy (Boedihardjo et al., 2023). The contribution there is analytical rather than mechanistic: improved covariance control strengthens the utility guarantee in an existing synthetic-data framework (Boedihardjo et al., 2023).

In noisy quantum optimization, the sampler based on two-qubit marginals provides a classical surrogate for noisy device output when the objective depends only on two-body correlations, such as Max-Cut or Ising/QUBO (Martinez et al., 29 Jul 2025). The cited paper emphasizes that the guarantee concerns expected objective values rather than total-variation closeness of the full output distributions, even though empirical energy histograms match closely (Martinez et al., 29 Jul 2025).

In energy disaggregation, semidefinite relaxation and randomized rounding are combined for inference in factorial hidden Markov models. The method samples from a Gaussian with mean equal to SDP marginals and covariance given by the SDP slack, then performs blockwise one-hot rounding and local repair to satisfy FHMM constraints (Shaloudegi et al., 2016). The source describes this as a Gaussian randomized rounding scheme that leverages the SDP’s second moments to construct feasible integer sequences (Shaloudegi et al., 2016).

6. Analytical themes, misconceptions, and limitations

Several misconceptions recur across these literatures. One is that Gaussian randomized rounding necessarily means independent Gaussian perturbation followed by thresholding. The sources show otherwise. Brownian rounding uses continuous martingale evolution with absorption (Sen, 2014), Grothendieck rounding uses multiple Gaussian hyperplanes and a sine post-processing (Boedihardjo et al., 2023), and heatmap regression uses randomized bilinear quantization with Gaussian labels (Yu et al., 2020). The underlying Gaussian object may be a stochastic process, a covariance matrix, a set of Gaussian directions, or merely a Gaussian-shaped label.

A second misconception is that the method always preserves the original relaxed marginals exactly. That is true in the Brownian packing formulation, where $$\max c^\top x \quad \text{subject to } A x \le b,\; x\in[0,1]^n,$$7 (Sen, 2014), and in the heatmap setting under the ideal decoder (Yu et al., 2020). It is not true in the Goemans–Williamson-style sign-rounding setting unless one performs an inverse-sine construction; the quantum-circuit paper explicitly distinguishes between a marginal-matching mapping and the GW-style mapping actually used in the analysis (Martinez et al., 29 Jul 2025).

A third point concerns feasibility. In packing ILPs, the Brownian procedure may terminate at a possibly infeasible point in $$\max c^\top x \quad \text{subject to } A x \le b,\; x\in[0,1]^n,$$8, so one still needs concentration, discrepancy control, repair, or a final Lovász Local Lemma stage to handle constraint violations (Sen, 2014, Madan et al., 2015). In PSD regularity and covariance-loss problems, by contrast, there is no combinatorial feasibility notion of the same type; the rounding objective is approximation of correlations and covariances rather than exact satisfaction of hard constraints (Boedihardjo et al., 2023).

The main limitations stated in the source material are likewise setting-specific. For Brownian packing methods, poor covariance choice can produce large variance along critical constraints, and maintaining projection operators or correlated Gaussian sampling can be more expensive than independent rounding (Sen, 2014). For PSD regularity, positive semidefiniteness is essential; Hadamard matrices rule out analogous Hilbert–Schmidt bounds without that assumption (Boedihardjo et al., 2023). For quantum sampling, two-body marginals can be insufficient for objectives involving higher-order correlations (Martinez et al., 29 Jul 2025). For heatmap regression, the method is lossless only in the ideal case; in practice accuracy remains limited by heatmap prediction quality and candidate-set selection (Yu et al., 2020).

These differences suggest that “Gaussian randomized rounding” is best regarded as a methodological template. A plausible implication is that its success depends less on Gaussianity alone than on the compatibility between Gaussian identities—martingale stopping, hyperplane separation, arcsine laws, subgaussian concentration—and the structural quantity one seeks to preserve.

7. Relation to classical randomized rounding and broader significance

Classical randomized rounding in the sense of Raghavan and Thompson rounds each variable independently according to its fractional LP value. The Brownian-motion papers present Gaussian randomized rounding as an alternate approach: it preserves expected objective value but replaces independence with a multidimensional random walk whose correlations can be used together with discrepancy arguments and the Lovász Local Lemma to exploit sparsity and structure (Sen, 2014, Madan et al., 2015). In random packing-row models, this leads to improved congestion guarantees over the worst-case independent-rounding bound (Madan et al., 2015).

In the SDP tradition, Gaussian randomized rounding is the canonical bridge from vector relaxations to discrete cuts or spins. The 2024 robust MAX-CUT paper stresses that the same randomization-projection framework preserves nominal approximation factors when passing to robust and distributionally robust counterparts, because the edgewise probability inequalities are weight-independent (Shi et al., 2024). The 2025 noisy-quantum work extends that viewpoint from approximation algorithms to surrogate sampling, showing that Gaussian sign rounding can replicate the behavior of noisy circuits for two-body objectives in a provable sense (Martinez et al., 29 Jul 2025).

In covariance and kernel approximation, the significance is different: Gaussian randomized rounding yields short, elementary proofs of nearly tight covariance-loss bounds and weak regularity lemmas for PSD matrices and kernels, with block approximants obtained by averaging over partition cells induced by Gaussian sign features (Boedihardjo et al., 2023). In that context, the method functions as a structural compression device rather than as a combinatorial optimizer.

Taken together, these works portray Gaussian randomized rounding as a versatile interface between continuous and discrete mathematics. In one direction it transforms LP or SDP solutions into integral objects while preserving expectation or approximation ratio; in another it converts continuous feature geometry into discrete partitions with controlled covariance loss; in yet another it encodes sub-pixel information into discrete heatmaps without information loss in the ideal model. The common architecture is the use of Gaussian randomness to expose analytically tractable symmetries—martingale, rotational, or subgaussian—that would be unavailable under deterministic thresholding or naive independent rounding (Sen, 2014, Boedihardjo et al., 2023, Shi et al., 2024).