Target-Phase Loss Function

• Target-Phase Loss Function is a family of loss objectives that use phase information to enforce consistency and adapt training.
• It encompasses both explicit phase comparisons, as in STFT-based speech enhancement, and implicit supervision via physically grounded surrogates.
• These loss functions are tailored to specific representations, yielding improved performance over magnitude-only approaches in various applications.

A target-phase loss function is not a single standardized object in the cited literature. Instead, it denotes a family of objectives in which supervision depends on phase information, phase consistency, target phases of a phase field, or the phase of optimization itself rather than on uniform pointwise reconstruction alone. Several of the cited papers explicitly state that they do not define a loss under that exact name, yet they introduce losses that are phase-aware, consistency-preserving, two-phase, or training-phase-adaptive. This suggests that the term is best treated as an umbrella label for objectives that either compare against a target phase, enforce valid phase structure, or adapt loss geometry to the phase relevant to the task (Ku et al., 2024, Sharifzadeh et al., 2023, Raymond et al., 2023).

1. Terminological scope and main variants

In the cited work, the word phase appears in at least three technically distinct senses. In time-frequency audio and binaural processing, phase is the argument of a complex STFT coefficient or a cross-channel phase difference. In phase-field implicit representations, phase denotes the binary states $u=-1$ and $u=1$. In online loss-function learning, phase refers to the phase of optimization, that is, early versus late training. A target-phase loss function therefore cannot be defined independently of representation.

Context	What acts as the target phase	Loss role
STFT speech enhancement	Clean phase or clean complex spectrum	Explicit phase-aware target matching
Binaural extraction	IPD or GCC-PHAT-derived delay structure	Spatial cue preservation
Ultrasound aberration correction	RF coherence and beamformed structure	Implicit phase-sensitive supervision
STFT consistency	Consistent complex spectrogram	Constraint instead of clean-phase regression
Implicit neural representations	Two target phases $u=-1,1$	Phase separation and interface regularization
Online loss learning	Current phase of training	Dynamic phase-dependent adaptation

The literature also draws a boundary between target-aware and target-phase objectives. The amplified-target loss for OCT segmentation is a target-dependent transformed-space loss that increases penalties in the central horizontal region of each B-scan, but it is not a phase loss in the signal-processing sense (Orlando et al., 2019). Likewise, some phase-retrieval papers contain the word phase in the task definition while still optimizing measurement-domain or intensity-domain objectives rather than a loss directly on phase (Li et al., 2018, Yazawa et al., 9 Apr 2026).

2. Explicit phase-sensitive supervision in spectral and binaural models

The clearest form of target-phase loss appears in STFT-domain speech enhancement. In the single-channel setting $X(k,n)=S(k,n)+N(k,n)$, the enhanced estimate is $\widehat S(k,n)=G(k,n)X(k,n)$, and the cited study restricts $G(k,n)$ to be real-valued. Because $\widehat S$ therefore keeps the noisy phase at inference, target phase enters only through the training loss. The phase-aware family includes complex spectral losses such as

$$\text{cMSE}=\left<|\widehat S-S|^2\right>, \qquad \text{cMAE}=\left<|\widehat S-S|\right>,$$

as well as the explicit phase-aware logarithmic loss

$$\text{PLSD} = \left< \left| \log_{10}\frac{\widehat A}{A} \right|^2 \left(2-\cos(\varphi_{\widehat S}-\varphi_S)\right) \right>.$$

The same paper also studies mixed objectives of the form

$$\mathcal L_{\text{mix}}=(1-\beta)\mathcal L_{\text{mag}}+\beta \mathcal L_{\text{complex}}.$$

Its main empirical conclusion is that combining magnitude-only with phase-aware objectives always leads to improvements, even when the phase is not enhanced, and that phase-sensitive improvement is best achieved by linear domain losses such as mean absolute error (Braun et al., 2020).

Binaural target sound extraction introduces a different but related notion of target phase. The interaural phase difference is defined from left and right STFT coefficients by

$u=1$0

and the corresponding IPD loss is

$u=1$1

The same work proposes a differentiable ITD-related surrogate based on GCC-PHAT cross-correlation,

$u=1$2

with loss

$u=1$3

The empirical contrast is decisive: the IPD loss reduces $u=1$4IPD from $u=1$5 rad to $u=1$6 rad, but worsens $u=1$7ITD-GCC from $u=1$8 to $u=1$9; the proposed ITD loss yields the best $u=-1,1$0ITD-GCC at $u=-1,1$1 while maintaining signal-level metrics (Hernandez-Olivan et al., 2024).

These results establish an important distinction. A target-phase loss can be explicitly phase-based and still optimize the wrong operational proxy. In the binaural case, minimizing IPD error does not guarantee preservation of the delay cue that GCC-PHAT actually estimates.

3. Implicit phase supervision through physically grounded surrogates

Not all target-phase losses compare angles or complex numbers directly. In ultrasound phase aberration correction without clean reference data, the network is trained in an aberration-to-aberration regime in which both input and target are randomly aberrated RF realizations of the same scene. The proposed objective is an adaptive mixed loss,

$u=-1,1$2

Here

$u=-1,1$3

The paper states that plain RF-domain MSE is inadequate because the network encounters challenges when comparing RF data directly, whereas the B-mode term stabilizes convergence and the later RF term restores fine phase-sensitive detail. For the simulated test image averaged over 100 aberrated versions, contrast improves from $u=-1,1$4 dB to $u=-1,1$5 dB, gCNR from $u=-1,1$6 to $u=-1,1$7, and speckle SNR is maintained from $u=-1,1$8 to $u=-1,1$9 (Sharifzadeh et al., 2023).

Synthetic-aperture radar moving-target imaging provides a second example of implicit phase supervision. The forward model embeds a motion-dependent phase history

$X(k,n)=S(k,n)+N(k,n)$0

and the central data-fidelity term penalizes

$X(k,n)=S(k,n)+N(k,n)$1

This is phase-aware because an incorrect target velocity produces an incorrect motion-induced phase $X(k,n)=S(k,n)+N(k,n)$2, which increases the residual. The loss therefore matches measured data to target-dependent phase histories rather than to a phase-angle label (Thammakhoune et al., 2021).

A plausible implication is that target-phase supervision need not appear as an explicit phase distance. It can instead arise from a forward operator whose residual is sensitive to the physically correct phase evolution.

4. Consistency-preserving and two-phase variational formulations

A major alternative to direct target-phase regression is to supervise validity rather than exact phase matching. In phase reconstruction and speech enhancement, the explicit consistency-preserving loss is defined on a complex spectrogram $X(k,n)=S(k,n)+N(k,n)$3 by

$X(k,n)=S(k,n)+N(k,n)$4

For phase reconstruction, the predicted spectrogram is $X(k,n)=S(k,n)+N(k,n)$5; for speech enhancement, it is $X(k,n)=S(k,n)+N(k,n)$6. The defining property is that the loss does not compare $X(k,n)=S(k,n)+N(k,n)$7 to a unique clean target phase. It only forces the predicted magnitude-phase pair to be a consistent STFT representation, that is, the spectrogram of a real signal. On VoiceBank-DEMAND phase reconstruction, the noisy-phase baseline obtains PESQ $X(k,n)=S(k,n)+N(k,n)$8, direct cosine phase loss gives $X(k,n)=S(k,n)+N(k,n)$9, anti-wrapping gives $\widehat S(k,n)=G(k,n)X(k,n)$0, and the consistency-preserving loss gives $\widehat S(k,n)=G(k,n)X(k,n)$1. In speech enhancement on WSJ0-CHiME3 with MetricGAN, the same loss achieves PESQ $\widehat S(k,n)=G(k,n)X(k,n)$2, compared with $\widehat S(k,n)=G(k,n)X(k,n)$3 for both cosine-plus-derivative and anti-wrapping-plus-derivative baselines (Ku et al., 2024).

The PHASE framework for implicit neural representations uses phase in a different sense: the target phases are the binary states $\widehat S(k,n)=G(k,n)X(k,n)$4 and $\widehat S(k,n)=G(k,n)X(k,n)$5, representing interior and exterior. Its central loss is

$\widehat S(k,n)=G(k,n)X(k,n)$6

where the double-well potential $\widehat S(k,n)=G(k,n)X(k,n)$7 has minima at the two target phases. The cited analysis shows that, under the stated asymptotic regime for $\widehat S(k,n)=G(k,n)X(k,n)$8, $\widehat S(k,n)=G(k,n)X(k,n)$9 $G(k,n)$0-converges to a sharp-interface functional whose minimizers satisfy proper occupancy, zero reconstruction error, and minimal perimeter. Under the special choice $G(k,n)$1, the transformed field

$G(k,n)$2

is tied to a viscous eikonal equation and converges, in the theorem’s stated sense, to a signed distance function (Lipman, 2021).

These two lines of work share a common structural move: the loss does not ask for one privileged phase realization. Instead, it constrains the output to lie on a valid phase manifold, whether that manifold is the set of consistent STFTs or the set of two-phase fields with controlled interfaces.

5. Target dependence, optimization phase, and adjacent formulations

The idea of aligning the loss with a relevant phase also appears in online loss-function learning. AdaLFL treats offline meta-learned losses as biased toward the very first few steps of training and updates the loss online after each base-model step: $G(k,n)$3

$G(k,n)$4

The paper frames this as a non-stationary bilevel optimization problem in which the loss itself evolves across training. On CIFAR-10 with AllCNN-C, ML$G(k,n)$5 with $G(k,n)$6 reaches $G(k,n)$7, while AdaLFL reaches $G(k,n)$8; increasing ML$G(k,n)$9’s inner horizon to $\widehat S$0 steps only improves to $\widehat S$1 (Raymond et al., 2023). Here the relevant target phase is not a physical phase variable but the current optimization phase. The loss is explicitly phase-dependent in time.

A related but distinct line is target-dependent transformed-space loss design. The amplified-target loss for photoreceptor segmentation is

$\widehat S$2

with a smoothed spatial weight matrix $\widehat S$3 that amplifies the central horizontal region of each OCT B-scan. The method is described as a target-dependent transformed-space loss, a spatially varying penalty scheme, and a prior-driven hard-region emphasis built on top of CE or MSE (Orlando et al., 2019). It is not a target-phase loss in the audio or phase-retrieval sense, but it is instructive because it shows how a loss can encode structural priors by transforming the target space before comparison.

This distinction matters. A target-dependent loss is not automatically a target-phase loss, and a phase-sensitive loss is not automatically tied to target transformation. The two design patterns intersect only when the transformed target or surrogate objective carries phase-relevant structure.

6. Common misconceptions, boundaries, and limitations

A recurrent misconception is that any paper about phase retrieval must contain an explicit target-phase loss. The cited evidence does not support that generalization. The supplementary material for robust phase retrieval via DC composite optimization states that there is no evidence of any explicit target-phase loss or loss directly on phase; the method appears to apply robust DC losses such as $\widehat S$4, capped $\widehat S$5, and trimmed $\widehat S$6 to phase-retrieval measurement residuals instead (Yazawa et al., 9 Apr 2026). The activated quartic objective for solving quadratic equations is likewise an intensity-domain loss,

$\widehat S$7

designed to eliminate spurious local minima in noiseless real-valued phase retrieval. In that setting the only local minimizers are $\widehat S$8, so the phase ambiguity is handled implicitly through the target equivalence class rather than through a direct loss on phase (Li et al., 2018).

Another misconception is that explicit phase matching is necessarily the best way to preserve the quantity of interest. The binaural TSE results show the opposite: IPD MSE sharply improves an IPD metric but degrades the ITD metric that better tracks arrival-time differences (Hernandez-Olivan et al., 2024). The STFT-consistency work makes a complementary point: exact phase regression can be worse than noisy phase, whereas enforcing consistency of the predicted magnitude-phase pair yields stronger speech reconstruction (Ku et al., 2024).

The cited work also makes clear that phase-aware supervision can be useful even when phase is not enhanced at inference. In single-channel speech enhancement with a real-valued suppression gain, the enhanced signal keeps noisy phase, yet combining magnitude-only with phase-aware objectives still improves performance (Braun et al., 2020). This suggests that target-phase information can act as a training-time regularizer on magnitude estimation under phase-constrained inference.

Finally, every formulation carries domain-specific assumptions. The amplified-target OCT loss assumes fovea-centered scans and centrally concentrated pathology (Orlando et al., 2019). The GCC-PHAT ITD loss assumes a bounded lag range and uses the entire 6-second signal as the transform window (Hernandez-Olivan et al., 2024). The explicit consistency-preserving loss depends on the chosen STFT window, hop size, and synthesis-analysis pair (Ku et al., 2024). PHASE depends on a double-well potential and small-interface asymptotics (Lipman, 2021). A target-phase loss function is therefore not merely a generic penalty on phase; it is a representation-bound objective whose meaning is fixed by the geometry, physics, or optimization regime of the underlying task.