HawkEye Loss Functions

• HawkEye Loss is a set of composite loss formulations with bounded, insensitive, and smooth properties that robustly handle outliers in regression and classification.
• It integrates components such as GAN loss, L1 reconstruction, and perceptual losses to enhance mmWave imaging and depth reconstruction.
• In video grounding, it combines cross-entropy and captioning losses to align temporal segments with descriptive text, improving model interpretability.

The term "HawkEye Loss" refers to multiple loss formulations introduced independently across distinct domains, each characterized by a bespoke composite or parametric design targeting robustness and fidelity in either supervised learning or multimodal alignment. This article surveys the main classes of HawkEye loss functions established in (i) robust regression and classification; (ii) mmWave imaging for self-driving; and (iii) temporal video grounding for video-text LLMs, with focus on their mathematical structure, key properties, and empirical context.

1. Definitions and Mathematical Formulations

Distinct but related HawkEye loss functions have been proposed. The parametric bounded HawkEye loss (often, "H-loss") is defined by the following piecewise formula for prediction residual $x$: $$\mathcal{L}_H(x) = \begin{cases} \lambda\left[1-(-a(x+\varepsilon)+1)e^{a(x+\varepsilon)}\right], & x \leq -\varepsilon \ 0, & |x| < \varepsilon \ \lambda\left[1-(a(x-\varepsilon)+1)e^{-a(x-\varepsilon)}\right], & x \geq \varepsilon \end{cases}$$ where $\varepsilon>0$ controls the insensitive (zero) zone, $a>0$ controls the curve's shape, and $\lambda>0$ is the upper bound on the loss. This formulation is established as the central technical innovation for both robust RVFL and SVR models (Akhtar et al., 2024, Akhtar et al., 2024).

For mmWave imaging, "HawkEye Loss" refers to a weighted sum: $$\mathcal{L}_{H}(G) = \mathcal{L}_{\mathrm{GAN}}(G,D^*) + \lambda_1 \mathcal{L}_1(G) + \lambda_p \mathcal{L}_p(G)$$ Here, $\mathcal{L}_{\mathrm{GAN}}$ is the standard conditional GAN loss, $\mathcal{L}_1$ is the mean absolute error, and $\mathcal{L}_p$ is a perceptual feature loss, each defined over 2D depth-map predictions from 3D mmWave heatmaps (Guan et al., 2019).

For temporal video grounding, HawkEye defines the loss as a sum of a cross-entropy grounding loss $L_{\text{ground}}$ and a sequence-level captioning loss $L_{\text{caption}}$: $$L_{\text{total}} = L_{\text{ground}} + \lambda L_{\text{caption}}$$ where $$L_{\text{ground}} = -\sum_{j=1}^{4} y_j \log \hat{p}_j$$, with $y$ a one-hot coarse time label and $\hat{p}$ the predicted probability over $\{\text{beginning, middle, end, throughout}\}$ (Wang et al., 2024).

2. Key Mathematical Properties

2.1 Robust Regression/Classification H-loss

• Boundedness: For all $x$, $0 \leq \mathcal{L}_H(x) \leq \lambda$. No large residual can contribute more than $\lambda$ to the loss, which confers outlier resistance.
• Insensitive Zone: By construction, $\mathcal{L}_H(x)=0$ for $|x|<\varepsilon$. Minor residuals are ignored, reducing overfitting to noise.
• Smoothness: The loss is $C^1(\mathbb{R})$ (continuously differentiable), with the first derivative

$$\mathcal{L}'_H(x) = \begin{cases} \lambda a^2 (x+\varepsilon) e^{a(x+\varepsilon)}, & x < -\varepsilon \ 0, & |x| \leq \varepsilon \ \lambda a^2 (x-\varepsilon) e^{-a(x-\varepsilon)}, & x > \varepsilon \end{cases}$$

Additionally, the loss is symmetric: $\mathcal{L}_H(x) = \mathcal{L}_H(-x)$, ensuring unbiased penalization of positive and negative errors (Akhtar et al., 2024, Akhtar et al., 2024).

2.2 Composite HawkEye Loss for Imaging

• Composite Structure: The loss aggregates adversarial fidelity (via a GAN objective), structural similarity (via $\ell_1$ reconstruction), and semantic similarity (via VGG-based perceptual features).
• No Explicit Insensitive Zone or Boundedness: The GAN and perceptual loss terms are not bounded by construction, but the use of $\ell_1$ instead of $\ell_2$ aids preservation of sharpness while controlling drift (Guan et al., 2019).

2.3 Temporal Video Grounding

• Coarse Temporal Targets: Replaces regression over timestamps with a 4-way classification loss, sidestepping difficulties involved in direct timestamp prediction.
• Combined Objectives: Adds sequence cross-entropy (for segment captioning) and categorical cross-entropy (for region grounding), ensuring alignment at both scene and text levels (Wang et al., 2024).

3. Theoretical and Practical Advantages

• Outlier Robustness: Boundedness in the regression/classification HawkEye loss ensures that outliers cannot dominate model gradients, directly mitigating model degradation in noisy settings.
• Sparsity and Regularization: The insensitive zone ignores small residuals, preventing over-adjustment to noise—a critical property in real-world, label-noisy datasets (Akhtar et al., 2024, Akhtar et al., 2024).
• Optimization Compatibility: Smoothness enables direct application of gradient-based optimizers (Adam, Nesterov Accelerated Gradient), ensuring stable convergence.
• Task-Specificity: In mmWave imaging, the composite loss is specifically tuned for enhancement of physical accuracy (geometry and structure) and suppression of modality artifacts (holes, ghosts), leveraging adversarial and perceptual synergies (Guan et al., 2019).
• Interpretability: The coarse-grained temporal classification encourages models to develop representational alignment at a timescale that is understandable and robust to test-time variations (Wang et al., 2024).

4. Comparative Analysis with Classical Loss Functions

A comparative table for regression/classification losses:

Loss	Bounded	Insensitive Zone	Smooth	Robust
Squared Error	No	No	Yes	No
$\varepsilon$-Insensitive	No	Yes	No	Partial
Huber	No	No	$C^1$	Partial
Ramp	Yes	Yes	No	Yes
Canal	Yes	Yes	No	Yes
Bounded-LS	Yes	No	Yes	Yes
HawkEye	Yes	Yes	Yes ($C^1$)	Yes

HawkEye is the first to combine boundedness, smoothness, and a symmetric insensitive zone in SVR/RVFL settings (Akhtar et al., 2024, Akhtar et al., 2024).

5. Integration into Learning Frameworks and Optimization

• Support Vector Regression (HE-LSSVR): HawkEye loss is substituted as the error penalizer in LSSVR, yielding a non-convex objective optimized by Adam. The kernelized objective is minimized directly: $$H(\alpha) = \tfrac{1}{2} \alpha^\top K\, \alpha + C \sum_{i=1}^N \ell_{\text{HE}}(y_i - (K\alpha)_i)$$ (Akhtar et al., 2024).
• Random Vector Functional Link Networks (H-RVFL): The HawkEye loss replaces MSE, optimized via Nesterov's accelerated gradient with time-decayed learning rates (Akhtar et al., 2024).
• GAN-based mmWave Imaging: The loss takes the form of a generator-discriminator minimax game with added $\ell_1$ (image-level) and $\ell_p$ (feature-level) penalties; hyperparameters $\lambda_1,\lambda_p$ are validated on held-out sets (Guan et al., 2019).
• Video-Text LLMs: The cross-entropy grounding task is coupled with captioning to form a total loss; negative–span cropping ensures true temporal localization learning (Wang et al., 2024).

6. Empirical Evaluation and Impact

• Robust SVR/RVFL: On UCI and KEEL benchmarks (up to 40 datasets, including artificially noisy scenarios), HE-LSSVR and H-RVFL consistently outperform SVM, classical SVR, LSSVR, and other robust baselines in both accuracy and stability. For example, H-RVFL achieves an average accuracy of 80.76% on small datasets and maintains robustness at high label-noise rates (e.g., 95.5% test accuracy in breast_cancer at 40% label noise) (Akhtar et al., 2024, Akhtar et al., 2024).
• mmWave Imaging: Composite HawkEye loss achieves 2.2–5.5× reductions in median dimension error, reduces specular holes (30%→13%), and ghosts (20%→3.7%) in depth reconstruction over classical cGAN or $\ell_2$ losses (Guan et al., 2019).
• Video-Text Grounding: HawkEye's 4-way coarse loss enables stable zero-shot [email protected] above 30% on Charades-STA (vs. ~14% for frame-level approaches), without loss of open-ended Q&A performance (Wang et al., 2024).

7. Extensions, Limitations, and Future Directions

• Non-convexity: All bounded, non-convex losses, including HawkEye, may converge only to local minima; no guarantee of global optimality is available (Akhtar et al., 2024, Akhtar et al., 2024).
• Computational Complexity: Bottlenecked by kernel or matrix-vector products in each iteration. Mini-batch and factorization methods could further reduce cost for large-scale problems.
• Generalization: The combination of boundedness and insensitivity aids generalization in high-noise and outlier regimes, suggesting applicability in emerging regression heads for deep neural architectures (Akhtar et al., 2024).
• Hyperparameter Selection: Proper choice of $(a,\lambda,\varepsilon)$ is required for different noise levels, requiring cross-validation or adaptive schemes.
• Applicability: The framework of HawkEye loss likely extends to other architectures (robust feature learning, semi-supervised loss design) and could motivate convergence theory for bounded, insensitive, smooth loss functions in non-convex optimization.

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In sum, HawkEye loss functions, despite their diverse instantiations across learning paradigms, are united by the pursuit of robust, stable, and semantically aligned model training through boundedness, insensitivity to noise, and smooth differentiability. They have established new baselines in both performance and interpretability across regression, classification, image synthesis, and multimodal temporal grounding tasks (Akhtar et al., 2024, Akhtar et al., 2024, Guan et al., 2019, Wang et al., 2024).