Novel Loss Functions

• Novel loss functions are objective formulations that embed domain-specific knowledge, such as spectral or physical constraints, to improve error minimization.
• They use techniques like frequency shaping, physics-informed constraints, and geometric regularization to enhance the performance of models in tasks like CT reconstruction and segmentation.
• Careful tuning of hyperparameters, such as balance factors in frequency or gradient losses, is crucial for achieving optimal stability and task-driven performance.

A novel loss function is an objective function designed to address specific shortcomings of standard loss formulations, often by encoding domain-specific knowledge, geometric or spectral properties, or direct physical and probabilistic constraints into the optimization objective of a machine learning model. Recent research across imaging, segmentation, regression, graph learning, and scientific computing demonstrates novel loss functions that surpass generic metrics (e.g., MSE, cross-entropy) by providing explicit control over which error modes are minimized, achieving task-driven model behavior and enhanced regularization.

1. Frequency-Shaped and Spectrally-Aware Losses in Imaging

Novel loss functions in the context of inverse problems like X-ray computed tomography (CT) often leverage spectral properties of the error. In "Design of Novel Loss Functions for Deep Learning in X-ray CT" (Rahman et al., 2023), the authors introduce two frequency-shaping loss terms:

• Low-frequency shaping loss $L_{\mathrm{freq}}$: By filtering the target-prediction error with a low-pass operator $f_1$ before applying the base norm (e.g., MSE), this loss forces the network to prioritize correction of low-to-mid spatial frequencies that account for global bias and streak artifacts in CT.

$$L_{\mathrm{freq}}(S_k) = \phi(f_1[X_k - g_\theta(Y_k)])$$

• High-frequency preservation loss $L_{\mathrm{pres}}$: By penalizing the removal of high-frequency structure from the input via a high-pass filter $f_2$, and scaling its influence by $\alpha$, the high-frequency loss encourages the retention of diagnostically relevant fine detail (e.g., bone edges).

$$L_{\mathrm{pres}}(T_k) = \phi(\alpha f_2[Y_k - g_\theta(Y_k)])$$

These terms are combined as:

$$L_k = \phi(f_1[X_k - g_\theta(Y_k)] + \alpha f_2[Y_k - g_\theta(Y_k)])$$

The overall training objective is a sum over all examples, with $\alpha$ controlling the trade-off between low-frequency correction (artifact suppression) and high-frequency preservation (texture fidelity). Adjusting $\alpha$ tunes the output image’s noise–resolution trade-off, aligning outputs to subjective radiological preferences. Empirically, mid-range values of $\alpha$ yield optimal spatial frequency balance and improved image homogeneity metrics (Rahman et al., 2023).

2. Domain-Matched and Physics-Informed Losses

Integrating physical acquisition principles directly into the loss objective ensures model outputs are quantitatively aligned with domain operations:

• Line-Integral Projection Loss (LIP-loss) in PET Attenuation Correction: In PET/CT, attenuation-corrected reconstructions depend on line integrals of the attenuation map, not direct image pixel values. The LIP-loss penalizes discrepancies in Radon projections over multiple angles between predicted and ground-truth attenuation maps, enforcing consistency with the underlying measurement physics (Shi et al., 2019):

$$LLIP(X, Y) = \frac{1}{N_{\text{angles}} N_{\text{bins}}} \sum_{k=1}^{N_{\text{angles}}}\sum_{i=1}^{N_{\text{bins}}}([P_k X]_i - [P_k Y]_i)^2$$

• Hough Space SR-Loss for Linear Structure Segmentation: For line-shaped targets such as contrails in satellite imagery, errors can be penalized in Hough space, where lines manifest as sharp peaks. SR Loss combines standard Dice loss in image space and an analogous Dice loss in discrete Hough space, directly enhancing geometric continuity and reducing fragmented predictions (Sun et al., 2023):

$$L_{\text{SR}}(P, G) = \alpha L_{\text{Dice}}(P, G) + (1-\alpha) L^{H}_{\text{Dice}}(H_P, H_G)$$

• Fokker–Planck-based Loss for Dynamic-Density Consistency: For systems described by stochastic differential equations, this loss enforces the steady-state Fokker–Planck equation locally using empirical or model-based densities and known drift fields (Lu et al., 24 Feb 2025):

$$\mathcal{L} = \sum_{\mathbf{x} \in \Omega} \Big| \mathbf{s}(\mathbf{x})^T \mathbf{f}(\mathbf{x}) + \nabla \cdot \mathbf{f}(\mathbf{x}) - D [ \mathbf{s}(\mathbf{x})^T \mathbf{s}(\mathbf{x}) + \nabla \cdot \mathbf{s}(\mathbf{x}) ] \Big|$$

These formulations systematically promote physically-warranted solutions and significantly improve domain-specific endpoints (e.g., PET attenuation fidelity, structure-preserving segmentation, parameter identification in dynamical systems).

3. Geometric, Shape, and Distribution-Regularized Losses

Some losses enforce geometric or distributional priors to address issues such as implausible segmentations or poor generalization:

• PCA-Driven Shape-Prior Loss: For organ segmentation, a principal-component analysis (PCA)-based shape loss penalizes deviations from the low-dimensional "shape space" of observed masks, typically via a Mahalanobis energy on projected coordinates (Karimzadeh et al., 2022):

$$L_{\text{shape}}(P) = \sum_{i=1}^{k} \frac{\alpha_i^2}{\lambda_i}$$

• Reduced Jeffries–Matusita (RJM) Loss: In classification, RJM is a bounded, convex alternative to cross-entropy, defined as the difference between unity and the square root of predicted probability for the true class. Its boundedness and lower Lipschitz constant yield demonstrably improved generalization (Lashkari et al., 13 Mar 2024):

$$\ell_{\text{RJM}}(\hat{y}, y) = \sum_{c=1}^C y_c (1 - \sqrt{\hat{y}_c})$$

• Hybrid/Composite Losses for Imbalanced or Structured Data: The HyTver loss combines a Tversky-index region term and a weighted cross-entropy, with hyperparameters controlling sensitivity to false positives versus false negatives, tuned for longitudinal multiple sclerosis lesion segmentation (Perera et al., 25 Aug 2025):

$$L_{\mathrm{HyTver}} = \gamma L_{\mathrm{mCE}} + (1 - \gamma) L_{\mathrm{Tversky}}$$

Hybridization yields superior Dice and surface distance metrics, and lower variability versus standard Dice and CE losses.

4. Losses for Optimization Stability, Conditioning, and Regularization

Optimization-based scientific ML and regularized learning benefit from losses that improve gradient flow and generalization:

• Stabilized Gradient Residual (SGR) Loss for PDE Solvers: SGR loss interpolates between mean-squared residual (MSE, prone to poor conditioning) and explicit gradient-matching. By tuning a parameter $\gamma$, SGR achieves improved conditioning, yielding up to three orders-of-magnitude convergence gains for PDE-discretized systems (Cao et al., 24 Jul 2025):

$$L_{\mathrm{SGR}}(q; \gamma) = (1-\gamma) \Vert f(q) \Vert_2^2 + \gamma f(q)^T (q - q_{\text{detach}}) + \gamma f_{\text{detach}}^T f_{\text{detach}}$$

• Lai Loss: Geometric Gradient Penalization: Introduces a pointwise multiplicative penalty based on the local prediction slope, acting as a spatially adaptive regularizer that controls smoothness and sensitivity in regression tasks (Lai, 13 May 2024):

$$e_{i,\mathrm{Lai}} = |\hat{y}_i - y_i| \cdot \omega(k_i, \lambda)$$

• Xtreme Margin Loss for Binary Classification: This loss introduces class-weighted nonlinear margin penalties, highly tunable via $\lambda_1$, $\lambda_2$ to directly target class-conditional precision, recall, or AUC by amplifying well-calibrated confidence for each class (Wali, 2022).

5. Graph and Combinatorial Optimization Losses

Addressing the lack of differentiable objectives in unsupervised combinatorial problems, novel losses make end-to-end training feasible:

• Differentiable Loss for Unsupervised Graph Partitioning in GNNs: Incorporates cut minimization, balance enforcement, and collapse avoidance, with all terms differentiable through soft assignments, enabling gradient-based training in NP-hard partition settings (Chaudhary, 2023):

$$L(\theta; G) = \alpha \ell_{\text{cuts}} + \beta \ell_{\text{balance}} + \gamma \ell_{\text{centrality}}$$

The GNN with this loss matches classic heuristics (e.g., Kernighan–Lin) on cut cost and greatly exceeds standard spectral methods in partition balance.

6. Evaluations, Generalization, and Practical Considerations

Novel loss functions are validated across quantitative metrics (accuracy, Dice, SSIM, NMAE, noise entropy, convergence speed), with empirical gains documented over baselines. Their success hinges on domain alignment (spectral, geometric, or physical), hyperparameter tuning, and, when applicable, computational efficiency (e.g., O(N) for hybrid segmentation losses, memory compression in Shadow Loss for Siamese/triplet networks (Khan et al., 2023)).

A caveat is that increased expressivity or domain specificity may require per-domain tuning or careful calibration to avoid instabilities or over-constraint, as in frequency-loss $\alpha$ selection or SGR’s $\gamma$ tuning for PDEs.

7. Future Directions and Generalization Potential

The consensus across recent literature is that principled, domain-tailored loss functions are key to unlocking superior generalization, robustness, and interpretability in both predictive and generative models. Many published strategies (e.g., frequency shaping, projection losses, physics-driven constraints, hybrid shapes) are directly extensible beyond their core domains, such as medical imaging, graph learning, or system identification, to other areas where structural prior knowledge or operational requirements can be encoded as differentiable objectives (Rahman et al., 2023, Shi et al., 2019, Sun et al., 2023, Chaudhary, 2023, Lu et al., 24 Feb 2025).

The development of novel loss functions remains an active research area, with promising directions including adaptive weighting schemes, integration of uncertainty quantification, and joint optimization with automated domain-adaptive tuning mechanisms.