X2 Loss Function: Robust Optimization Technique

• X2 Loss Function is a family of modified quadratic losses that incorporate activation or weighting mechanisms for robust and adaptive optimization.
• It modulates squared residuals using activation functions or proportional weights to manage tail behavior and enhance fit accuracy in applications such as EIS and deep metric learning.
• The adaptive formulation improves convergence and parameter estimation while trading off computational efficiency for higher quality fit results.

The X2 loss function refers to a family of modified loss formulations in which the classical quadratic loss is coupled with either proportional weighting, activation-based truncation, or nonlinear logits, depending on problem context and application. Its major theoretical and practical motivation is to achieve robust optimization landscapes, enhanced fit accuracy, or adaptive separation in settings as diverse as nonconvex inverse problems, deep metric learning, and nonlinear least squares estimation.

1. Technical Formulation and Variants

The core structural principle of X2 loss is the modulation of the standard squared residual, either by activation functions or proportional weights that regularize tail behavior or adapt to measurement structure.

In the real-valued quadratic equation setting (Li et al., 2018), the X2 loss is constructed as:

$$f(z) = \frac{1}{2m} \sum_k \Big[ \big( (a_k^\top z)^2 - (a_k^\top x)^2 \big)^2 \cdot h(|a_k^\top z|^2/\|z\|^2) \cdot h(m|a_k^\top x|^2/\|y\|_1) \Big]$$

where $h(u)$ is an activation function defined via bounding thresholds $\beta$ and $\gamma$:

$$h(u) = \begin{cases} 1, & 0 \leq u \leq \beta \ \in [0,1], & \beta < u < \gamma \ 0, & u \geq \gamma \end{cases}$$

For nonlinear least squares estimation in electrochemical impedance spectroscopy (EIS) (Jaberi et al., 7 Oct 2025), the X2 loss realizes proportional weighting:

$$S_{X_2} = \sum_{i=1}^n \frac{[Z_{t,i} - Z_m(\omega_i, P)]^2}{|Z_{t,i}|^2}$$

where $Z_{t,i}$ is the true impedance at the $i$th frequency, and $Z_m(\omega_i, P)$ is the ECM model prediction.

In adaptive angular margin learning for face recognition (Xu et al., 2023), the X2 loss takes the form of a quadratic function applied to angles:

$f_R(\theta) = a \cdot (\theta - h)^2 + k$

where $a < 0$ dictates curvature, $h$ shifts the vertex, and $k$ sets vertical displacement. Here, the quadratic provides a margin that grows monotonically with inter-class angle.

2. Geometric Landscape and Theoretical Guarantees

The X2 loss formulation for solving real-valued quadratic equations (Li et al., 2018) is rigorously proved to eliminate spurious local minima when measurements are Gaussian and the sample complexity $m \gtrsim n$. The only local minimizers of $f(z)$ are at $\pm x$, where $x$ is the true signal.

Critical points not corresponding to global solutions are characterized by negative directional curvature: for every non-global stationary point, there exists a direction along which the Hessian possesses a strictly negative eigenvalue. This property implies that descent-based or Hessian-aware optimization strategies will not become trapped at undesirable stationary points.

Activation functions in the X2 construction play a pivotal role, truncating heavy-tailed gradient and Hessian contributions and ensuring the loss landscape is free of pathological minima caused by outlier measurements or non-Gaussian noise.

3. Adaptive Margins and Feature Separability

In deep metric learning tasks such as face recognition, the X2 loss (X2-Softmax variant) adaptively enforces angular margins according to actual inter-class separations (Xu et al., 2023). Unlike fixed-margin losses (e.g., CosFace or ArcFace), the quadratic mapping induces a margin $\Delta\theta$ that increases as the angle between class weights grows:

$$\Delta\theta = \arccos(a \cdot (\theta_1 - h)^2 + k) - \theta_1$$

Experimental results demonstrate that X2-Softmax leads to tighter intra-class feature clusters and more robust separation, with smaller "confusion regions" in the cosine similarity distributions of positive and negative pairs. The adaptive margin yields higher accuracy and more consistent true accept rates (TAR) at low false accept rates (FAR) across diverse face recognition benchmarks.

4. Weighted Residuals for Model Fitting

In nonlinear model fitting for EIS (Jaberi et al., 7 Oct 2025), the X2 loss's proportional weighting normalizes residuals across frequency regimes, preventing regions with large impedance from dominating parameter estimation. This weighted approach achieves the highest R2 scores (overall, magnitude, and phase), lowest chi-squared values, and excellent convergence rates (about 98.7% of fits reach the target threshold).

Comparative analysis shows that while log-B loss functions can yield nearly comparable accuracy at lower computational cost (1.4× faster per fit), the X2 loss is optimal when the highest fit quality is paramount, particularly for scientific inference applications.

5. Applications Across Domains

• Phase Retrieval & Quadratic Inverse Problems: X2 loss frameworks are specifically constructed to address phase retrieval problems arising in X-ray crystallography, electron microscopy, and optical imaging, as well as low-rank matrix recovery. Their favorable landscapes allow guarantees of global optimality under minimal sample complexity (Li et al., 2018).
• Face Recognition & Metric Learning: The adaptive margin feature of the X2 loss (X2-Softmax) yields state-of-the-art performance in large-scale face recognition, especially under class imbalance and varied inter-class similarity (Xu et al., 2023).
• Spectroscopy & Model Parameterization: For circuit model fitting from EIS data, X2 loss improves reliability of parameter estimates and fit quality, especially where a high-quality fit supersedes computational efficiency (Jaberi et al., 7 Oct 2025).

6. Computational Considerations and Tradeoffs

There is a documented tradeoff between fit quality and computational efficiency. The X2 loss often incurs higher computational cost—for EIS circuit model fitting, averaging 1.45 s per fit—compared to alternatives such as log-B (1.05 s per fit), which may be preferable for large-scale, time-sensitive applications.

Conversely, the X2-Softmax loss for face recognition is computationally efficient due to its quadratic form, obviating the need for costly hyperparameter tuning or careful selection of fixed margins. In scenarios that demand both adaptivity and computational parsimony, ongoing research seeks further parameterizations of the X2 framework or extensions to higher-order functions.

7. Extensions and Implications

Theoretically, the technique of coupling a smooth loss (quadratic) with activation or truncation mechanisms might be generalized to other inverse problems where heavy-tailed data distributions or outlier contamination affect optimization geometry. Extensions to low-rank matrix recovery or complex-valued measurements have been suggested (Li et al., 2018).

A plausible implication is that the modularity of X2 constructions—by adjusting activation thresholds, weighting schemes, or polynomial exponents—could facilitate tailored loss functions for a wide array of data-driven scientific or engineering applications, combining robust optimization geometry with practical adaptability.