First-Order Approach: Methods and Applications

• First-Order Approach (FOA) is a versatile strategy that replaces complex, higher-order systems with primary, first-order variables across various domains.
• It reformulates challenging global or high-dimensional problems into manageable representations in optimization, moral-hazard economics, spatial audio, and numerical PDEs.
• Its practical strengths include improved computational efficiency and clearer structural insights, though limitations emerge in settings like shifting-support conditions and coarse spatial resolutions.

In the works considered here, “First-Order Approach (FOA)” denotes several distinct but structurally related research programs built around first-order objects: first-order information in optimization, first-order conditions in contract theory, first-order reformulations of higher-order field equations, and, in spatial audio, First-Order Ambisonics as a four-channel spherical-harmonic representation. Across these literatures, the common move is to replace a more difficult global, higher-order, or high-dimensional problem by a representation in which first-order variables, first-order constraints, or first-order signal components become the primary design objects (Guille-Escuret et al., 2020, Jiang, 18 Sep 2025, Chamarthi et al., 2019, Zlosnik et al., 2016, Mazzon et al., 2019).

1. Cross-domain meaning

In first-order optimization, the term refers to deterministic methods that use function values and gradients, for example

$$x_{n+1}= \mathcal{A}_\theta\Big( \{x_{i}\}_{i=0\ldots n}, \{f(x_i)\}_{i=0\ldots n}, \{\nabla f(x_i)\}_{i=0\ldots n} \Big),$$

and to theories that tune such methods by condition measures rather than second-order structure. In moral-hazard economics, FOA is the standard relaxation replacing the global incentive-compatibility constraint by the agent’s first-order condition. In anisotropic diffusion, FOA means rewriting a second-order elliptic problem as a first-order hyperbolic system in pseudo-time. In conformal gauge gravity, it denotes a polynomial $SU(2,2)$ theory whose fundamental variables satisfy first-order field equations. In spatial audio, the same acronym most often denotes First-Order Ambisonics, where the sound field is truncated to four channels and then processed, decoded, compressed, or upscaled by FOA-specific algorithms (Azevedo et al., 23 Jun 2025, Jiang, 18 Sep 2025, Chamarthi et al., 2019, Zlosnik et al., 2016, Milstein et al., 30 Sep 2025).

These usages are not interchangeable. In some fields FOA is an approximation or relaxation; in others it is a representation format; in others it is a claim about the differential order of the governing equations. A plausible implication is that “first-order” functions less as a single doctrine than as a recurring methodological strategy for isolating the variables or constraints regarded as operationally fundamental.

2. First-order optimization

A broad optimization-theoretic formulation studies FOA as the class of continuous deterministic methods driven by $(x_i,f(x_i),\nabla f(x_i))$ histories, and asks which condition measures are stable under perturbations actually relevant to algorithmic trajectories. One such perturbation model introduces

$$\|h\|_*= \sup_{x\in\mathbb{R}^d\backslash X^*}\frac{\|\nabla h(x)\|_2}{d(x,X^*)},$$

and proves that small $\|\cdot\|_*$-perturbations induce small finite-time perturbations in the behavior of any continuous FOA. Against that background, smoothness and strong convexity are shown to be “continuous nowhere,” whereas alternatives such as $PL^\pm$, $EB^\pm$, $RSI^\pm$, $QG^\pm$, and ${}^*\!SC^\pm$ are continuous in the required sense. The paper then derives Gradient Descent rates under these alternative upper/lower condition pairs and argues that robust tuning should use condition numbers built from continuous metrics rather than unstable curvature bounds (Guille-Escuret et al., 2020).

A different line of work treats FOA on Riemannian manifolds. For composite problems

$SU(2,2)$0

the proposed manifold FOA replaces Euclidean extrapolation by lifting and retraction. Its local model is

$SU(2,2)$1

with accelerated update

$SU(2,2)$2

The proved rate is

$SU(2,2)$3

so the paper’s “quadratic convergence” is an $SU(2,2)$4 objective rate in the FISTA/Nesterov sense rather than Newton-type local quadratic convergence (Chen et al., 2015).

The complementary negative theory asks when no worst-case convergence theorem can exist. For stationary first-order methods,

$SU(2,2)$5

the search for cyclic trajectories is reduced to a finite-dimensional SDP through interpolation conditions. A $SU(2,2)$6-cycle is certified by minimizing

$SU(2,2)$7

This constructive program identifies broad nonconvergent regions for heavy-ball, boundary-cycle behavior for constant-parameter Nesterov acceleration, and essentially tight stability regions for inexact gradient descent, thereby complementing Lyapunov-based sufficient conditions with explicit impossibility certificates (Goujaud et al., 2023).

3. Moral hazard and principal–agent theory

In continuous-time moral-hazard terminology, FOA is the standard replacement of the agent’s global IC constraint by a local stationarity condition. With effort $SU(2,2)$8, contract $SU(2,2)$9, and agent utility

$(x_i,f(x_i),\nabla f(x_i))$0

the original principal’s problem imposes

$(x_i,f(x_i),\nabla f(x_i))$1

whereas the FOA-relaxed problem replaces this by

$(x_i,f(x_i),\nabla f(x_i))$2

In the high-reservation-utility analysis, the relaxed solution has the canonical form

$(x_i,f(x_i),\nabla f(x_i))$3

or

$(x_i,f(x_i),\nabla f(x_i))$4

The main theorem states that FOA is valid for $(x_i,f(x_i),\nabla f(x_i))$5 for any sufficiently high reservation utility $(x_i,f(x_i),\nabla f(x_i))$6, and under log utility the optimal contracts become option contracts for Gaussian, exponential, binomial, Gamma, and Laplace output models (Azevedo et al., 23 Jun 2025).

That validity is not uniform. When the support of output shifts with effort, FOA can fail even under strict unimodality because the agent’s expected utility may have an interior kink optimum. In the additive-noise example

$(x_i,f(x_i),\nabla f(x_i))$7

the quota-bonus contract

$(x_i,f(x_i),\nabla f(x_i))$8

implements the first-best effort $(x_i,f(x_i),\nabla f(x_i))$9 with $$\|h\|_*= \sup_{x\in\mathbb{R}^d\backslash X^*}\frac{\|\nabla h(x)\|_2}{d(x,X^*)},$$0, while the FOA-relaxed problem excludes quota-bonus contracts altogether. The paper therefore argues that FOA is not a valid relaxation in shifting-support settings and proposes the Implementation Relaxation Approach (IRA), which relaxes the implementable effort-utility set rather than the IC constraint itself (Jiang, 18 Sep 2025).

A recurrent misconception is that unimodality alone rescues FOA. The shifting-support analysis shows that this is false once the optimum is non-differentiable: global IC may hold while the first-order condition does not.

4. First-Order Ambisonics as a first-order spatial representation

In spatial audio, FOA ordinarily means First-Order Ambisonics: a four-channel sound-field representation, commonly written as $$\|h\|_*= \sup_{x\in\mathbb{R}^d\backslash X^*}\frac{\|\nabla h(x)\|_2}{d(x,X^*)},$$1, with one omnidirectional channel and three first-order directional channels. In the DCASE convention used for FOA-domain augmentation, the steering responses are

$$\|h\|_*= \sup_{x\in\mathbb{R}^d\backslash X^*}\frac{\|\nabla h(x)\|_2}{d(x,X^*)},$$2

so the directional channels behave like the Cartesian direction vector $$\|h\|_*= \sup_{x\in\mathbb{R}^d\backslash X^*}\frac{\|\nabla h(x)\|_2}{d(x,X^*)},$$3. FOA is attractive because it uses four channels and can be captured with only four microphones, but the same low order causes reduced externalization, poor spatial resolution, and coarse angular selectivity in binaural and loudspeaker rendering (Mazzon et al., 2019, Berebi et al., 2023, Milstein et al., 30 Sep 2025).

That vector structure makes FOA particularly suitable for data augmentation in direction-of-arrival estimation. Rotations and reflections can be applied directly to $$\|h\|_*= \sup_{x\in\mathbb{R}^d\backslash X^*}\frac{\|\nabla h(x)\|_2}{d(x,X^*)},$$4 while the labels are transformed analytically. The literature surveyed here gives three augmentation families—“16 Patterns,” “Labels First,” and “Channels First”—and reports DOA-error improvements of around 40% across two DCASE2019 systems, with the gains depending strongly on whether the transformed labels remain close to the dataset’s original angle support (Mazzon et al., 2019).

In headphone reproduction, FOA also motivates order-specific HRTF preprocessing. For FOA binaural rendering, the left/right ear signals are

$$\|h\|_*= \sup_{x\in\mathbb{R}^d\backslash X^*}\frac{\|\nabla h(x)\|_2}{d(x,X^*)},$$5

with $$\|h\|_*= \sup_{x\in\mathbb{R}^d\backslash X^*}\frac{\|\nabla h(x)\|_2}{d(x,X^*)},$$6 for FOA. The iMagLS method replaces pure magnitude least squares by a joint magnitude-plus-ILD objective,

$$\|h\|_*= \sup_{x\in\mathbb{R}^d\backslash X^*}\frac{\|\nabla h(x)\|_2}{d(x,X^*)},$$7

optimized by BFGS over $$\|h\|_*= \sup_{x\in\mathbb{R}^d\backslash X^*}\frac{\|\nabla h(x)\|_2}{d(x,X^*)},$$8 kHz. On simulated KEMAR, it preserves horizontal-plane ILD error below $$\|h\|_*= \sup_{x\in\mathbb{R}^d\backslash X^*}\frac{\|\nabla h(x)\|_2}{d(x,X^*)},$$9 dB for most incident angles while increasing magnitude error by only $\|\cdot\|_*$0 dB on average above $\|\cdot\|_*$1 kHz, showing that FOA-specific HRTF optimization can target binaural cue preservation rather than only per-ear spectral fidelity (Berebi et al., 2023).

5. Learned FOA enhancement, coding, and physically structured modeling

One major FOA research direction is order enhancement. DiffAU treats FOA-to-HOA conversion as conditional generative sampling, learning $\|\cdot\|_*$2 for missing higher-order channels given the observed FOA channels. For the $\|\cdot\|_*$3 to $\|\cdot\|_*$4 task, the input is a 4-channel FOA signal and the output a 16-channel third-order Ambisonics signal, generated by two cascaded diffusion blocks $\|\cdot\|_*$5 and $\|\cdot\|_*$6. In anechoic 1–4 speaker scenes, the reported STFT-SDR on the predicted HOA channels is $\|\cdot\|_*$7 dB overall versus $\|\cdot\|_*$8 dB for a compressed-sensing baseline, and the MUSHRA-style listening test finds the upscaled result essentially indistinguishable from true third-order Ambisonics in the tested single-speaker conditions (Milstein et al., 30 Sep 2025). A waveform-domain alternative based on Conv-TasNet predicts full HOA3 from FOA and reports positional error $\|\cdot\|_*$9 dB versus $PL^\pm$0 dB for native HOA3 and $PL^\pm$1 dB for conventional FOA rendering, together with an 80% median perceived-quality improvement over traditional FOA rendering (Nawfal et al., 1 Aug 2025). At the steering-vector level, SIRUP uses a VAE plus latent diffusion to upmix FOA steering vectors to HOA steering vectors, improving directivity index, beamwidth, sidelobe suppression, source localization, and beamforming-based speech denoising relative to FOA baselines (Picard et al., 18 Feb 2026).

FOA has also become a target for ultra-low-bitrate coding. The FOA Tokenizer extends WavTokenizer to four-channel FOA at 24 kHz and adds a spatial consistency loss based on time-frequency intensity-vector alignment,

$PL^\pm$2

with the cosine term computed from input and reconstructed FOA intensity vectors. The codec compresses FOA into 75 discrete tokens per second, corresponding to $PL^\pm$3 kbps, and reports mean angular errors of $PL^\pm$4, $PL^\pm$5, and $PL^\pm$6 on simulated reverberant mixtures, clean non-reverberant speech, and real-RIR mixtures, respectively. An ablation without the spatial consistency loss reaches catastrophic angular error $PL^\pm$7, even though acoustic scores remain similar, making explicit that waveform fidelity alone does not preserve FOA directional structure (Sudarsanam et al., 25 Oct 2025).

Another line of work imposes physical structure on FOA room-impulse-response modeling. A physics-informed neural field for FOA RIRs identifies $PL^\pm$8 with pressure and $PL^\pm$9 with scaled negative particle velocity, using the momentum and continuity priors

$EB^\pm$0

and shows that FOA-specific priors outperform a $EB^\pm$1-only wave-equation prior for early RIR interpolation (Masuyama et al., 9 Jul 2025). A later reformulation replaces direct four-channel prediction by a scalar velocity-potential neural field $EB^\pm$2, from which FOA is recovered as

$EB^\pm$3

This makes the linearized momentum equation hold exactly by construction and improves FOA RIR reconstruction, especially in sparse-measurement and boundary-only regimes (Masuyama et al., 23 Mar 2026).

FOA has also become an explicit multimodal representation. Spatial-Omni feeds the $EB^\pm$4 channel to a pretrained audio encoder while a dedicated SO-Encoder processes 4 FOA mel channels plus 3 intensity-vector channels and produces 10 Hz spatial latents, later compressed to 2.5 Hz spatial tokens for an Omni LLM. The resulting SO-Dataset contains about 400K FOA clips and 2.1M spatial QA pairs, and the model improves spatial reasoning, azimuth/elevation estimation, distance estimation, multi-hop reasoning, and direction-conditioned speech understanding relative to monaural baselines (Zhu et al., 9 Jun 2026). In video generation, DynFOA conditions latent FOA diffusion on dynamic source tracking, depth, semantics, and 3D Gaussian Splatting geometry/material cues, then rotates the generated FOA under listener head motion. On the reported benchmarks, it consistently improves DOA, EDT, FD, KL, STFT, SI-SDR, and MOS metrics over prior visual-to-spatial-audio baselines (Luo et al., 3 Apr 2026).

6. First-order hyperbolic reformulation of anisotropic diffusion

In numerical PDEs, FOA denotes a first-order hyperbolic system method for the anisotropic diffusion equation. Starting from

$EB^\pm$5

the method introduces auxiliary gradients

$EB^\pm$6

and evolves the pseudo-time system

$EB^\pm$7

The preconditioned flux Jacobian has real eigenvalues, so the pseudo-time problem is hyperbolic and can be discretized with upwind high-order finite differences. The key design parameter is the relaxation time

$EB^\pm$8

derived from a Fourier analysis and an optimal length scale. With this choice, the reported fifth-order schemes remain uniformly accurate as the anisotropy ratio rises to $EB^\pm$9, and $RSI^\pm$0, $RSI^\pm$1, and $RSI^\pm$2 are computed simultaneously to the same formal order (Chamarthi et al., 2019).

The point of the first-order reformulation is therefore not merely algebraic. It supplies a hyperbolic pseudo-dynamics in which error modes propagate, supports standard upwind/compact reconstructions, and makes anisotropy-independent high-order accuracy straightforward once the preconditioner is scaled appropriately.

7. First-order gauge gravity and conformal structure

In gauge-gravity theory, FOA denotes a first-order $RSI^\pm$3 formulation of gravity based on a gauge field $RSI^\pm$4 and an adjoint Higgs field $RSI^\pm$5. The symmetry breaking

$RSI^\pm$6

is driven by $RSI^\pm$7, and the action is polynomial in $RSI^\pm$8, $RSI^\pm$9, $QG^\pm$0, and $QG^\pm$1: $QG^\pm$2 Because the fundamental variables are $QG^\pm$3 and $QG^\pm$4, the corresponding field equations are first-order in derivatives. After symmetry breaking, the residual $QG^\pm$5 acts as a local scale symmetry, and the broken connection decomposes into a Lorentz spin connection, an $QG^\pm$6 connection, and two frame fields $QG^\pm$7 and $QG^\pm$8 that transform with opposite local weights (Zlosnik et al., 2016).

Within constrained sectors of the theory, this first-order structure reproduces both conformalized GR and fourth-order Weyl gravity. When the two frames align as

$QG^\pm$9

the scale connection becomes pure gauge on-shell and the theory reduces to a conformally coupled scalar-tensor action, which in the gauge ${}^*\!SC^\pm$0 becomes Einstein gravity. In a different constrained sector, algebraically eliminating one frame yields a Weyl-squared action, so ordinary fourth-order conformal gravity appears as a derived limit of the underlying first-order polynomial theory. Around maximally symmetric solutions, the perturbation spectrum contains the usual graviton plus a massive scalar and a massless one-form, and the matter sector can be written in fully gauge-invariant polynomial form with first-order field equations and no auxiliary fields (Zlosnik et al., 2016).

Taken together, these literatures show that “First-Order Approach” names a recurring strategy rather than a single formalism. In some domains the first-order structure is a relaxation that can fail or become valid only under additional hypotheses; in others it is a representation or gauge principle that reorganizes the entire theory. The technical consequences depend on the domain, but the recurring theme is that first-order variables are treated as the natural interface between structure, computation, and observables.