Normalized Spatiotemporal Gradient (NSG)

• NSG is a spatiotemporal statistic defined as the ratio of spatial probability gradients to temporal density changes, capturing plausible video dynamics.
• It leverages pre-trained diffusion models to estimate spatial gradients and uses motion-aware approximations for temporal changes under continuity assumptions.
• Empirical results show that integrating NSG into detection frameworks improves AI-generated video detection through enhanced recall and F1 scores via MMD analysis.

Searching arXiv for the specified paper and directly related uses of “Normalized Spatiotemporal Gradient.” Normalized Spatiotemporal Gradient (NSG) is a statistic introduced for AI-generated video detection within a physics-driven modeling framework that treats video evolution through a probability-flow perspective. It is defined as a ratio of spatial probability gradients to temporal density changes, with the aim of capturing whether a video follows plausible spatiotemporal dynamics under a probability-conservation-style constraint. In the formulation proposed in "Physics-Driven Spatiotemporal Modeling for AI-Generated Video Detection," NSG serves as the central feature representation for the detector NSG-VD, which compares NSG features from test videos against those from real videos via Maximum Mean Discrepancy (MMD) (Zhang et al., 9 Oct 2025).

1. Conceptual definition

NSG is designed to quantify how a video’s probability density evolves over time while retaining information about the spatial structure of each frame. Its defining expression is

$$\mathbf{g}(\mathbf{x}, t) = \frac{\nabla_{\mathbf{x}} \log p(\mathbf{x}, t)}{-\partial_t \log p(\mathbf{x}, t)+\lambda},$$

where $\mathbf{g}(\mathbf{x}, t)$ is the NSG statistic at time $t$, $\nabla_{\mathbf{x}} \log p(\mathbf{x}, t)$ is the spatial probability gradient, $-\partial_t \log p(\mathbf{x}, t)$ is the temporal density change, and $\lambda>0$ is a stabilizer to prevent division by very small values (Zhang et al., 9 Oct 2025).

The intended interpretation is explicit: NSG is a ratio of spatial probability gradients to temporal density changes. In this construction, the numerator acts as a score-like measure of spatial sensitivity of the video distribution, whereas the denominator measures how the log-density changes over time. This coupling is meant to characterize whether spatial structure and temporal evolution remain aligned in a physically consistent way.

The motivating claim is that natural videos should obey a continuity-like spatiotemporal dynamic, while AI-generated videos may violate these dynamics in subtle ways even when individual frames appear visually realistic. This suggests that NSG is not aimed at identifying overt rendering artifacts; rather, it targets deviations in the underlying evolution of video probability mass.

2. Derivation from probability flow conservation

The NSG construction begins from a probability-flow view of video evolution. Let $p(\mathbf{x}, t)$ denote the probability density of a video state $\mathbf{x}$ at time $t$, and let $\mathbf{v}(\mathbf{x}, t)$ be a probability-flow velocity field. The probability-flow density is written as

$\mathbf{g}(\mathbf{x}, t)$0

Assuming conservation of probability mass, the density satisfies the continuity equation

$\mathbf{g}(\mathbf{x}, t)$1

After substituting $\mathbf{g}(\mathbf{x}, t)$2, dividing by $\mathbf{g}(\mathbf{x}, t)$3, and applying the chain rule, the resulting relation is

$\mathbf{g}(\mathbf{x}, t)$4

The derivation then introduces a simplifying physical assumption: in smoothly varying video distributions, the divergence term $\mathbf{g}(\mathbf{x}, t)$5 is subdominant. Under this approximation,

$\mathbf{g}(\mathbf{x}, t)$6

Because $\mathbf{g}(\mathbf{x}, t)$7 is not uniquely recoverable, the relation is normalized, producing NSG as the operational statistic (Zhang et al., 9 Oct 2025).

The derivation therefore depends on four stated assumptions or constraints: smooth spatiotemporal evolution, a subdominant divergence term in the continuity equation, brightness constancy or small inter-frame motion for temporal approximation, and stabilization via $\mathbf{g}(\mathbf{x}, t)$8. A plausible implication is that NSG is best understood as an approximation rooted in a physically motivated continuum model rather than as an exact observable of raw video sequences.

3. Estimation with pre-trained diffusion models

Direct computation of NSG requires the unknown quantities $\mathbf{g}(\mathbf{x}, t)$9 and $t$0. The proposed estimator approximates these terms using pre-trained diffusion models and motion-aware temporal modeling (Zhang et al., 9 Oct 2025).

For the spatial term, a pre-trained score network $t$1 is used to estimate the gradient of the log-density:

$t$2

where $t$3 is the $t$4-th frame and $t$5 is the learned score model from a pre-trained diffusion model.

For the temporal term, the method adopts a brightness-constancy-style assumption from optical flow theory,

$t$6

for small $t$7 and $t$8. A Taylor expansion then yields the approximation

$t$9

This uses inter-frame displacement to approximate temporal density change without explicitly computing full optical flow. Combining the two approximations produces the practical estimator

$\nabla_{\mathbf{x}} \log p(\mathbf{x}, t)$0

The paper characterizes this estimator as using diffusion-model scores for spatial gradients, frame-to-frame motion to estimate temporal density change, avoiding explicit flow inversion, and preserving the physical continuity motivation. In that sense, NSG is neither a purely appearance-based descriptor nor a conventional motion descriptor; it is a hybrid statistic whose numerator and denominator are both tied to an underlying probability model.

4. Role in NSG-VD

NSG is used in the full detection method NSG-VD by first constructing a sequence of NSG features across all frames of a video:

$\nabla_{\mathbf{x}} \log p(\mathbf{x}, t)$1

This sequence becomes the feature representation of the video (Zhang et al., 9 Oct 2025).

Detection is then based on Maximum Mean Discrepancy between NSG features from a reference set of real videos and NSG features from the test video. The method uses an unbiased MMD estimator with a kernel $\nabla_{\mathbf{x}} \log p(\mathbf{x}, t)$2 operating on NSG features, and a deep kernel is used to improve discriminative power. The decision rule is binary: the video is classified as fake if MMD $\nabla_{\mathbf{x}} \log p(\mathbf{x}, t)$3, and real otherwise.

Within this architecture, NSG provides the physically grounded representation, while MMD serves as the statistical test on distributions in reproducing-kernel Hilbert space. The stated motivation is that many existing detectors emphasize surface artifacts, appearance inconsistencies, or explicit motion features, whereas NSG-VD compares videos in a feature space intended to reflect probability-flow-consistent spatiotemporal behavior.

This design also clarifies an important interpretive point. NSG by itself is not the detector; it is the feature statistic. The detector is NSG-VD, which operationalizes NSG features through distribution comparison against real-video references.

5. Theoretical separation properties

The paper derives an upper bound on NSG feature distances between real and generated videos under a Gaussian modeling assumption. Real videos are assumed to follow

$\nabla_{\mathbf{x}} \log p(\mathbf{x}, t)$4

while generated videos are modeled as

$\nabla_{\mathbf{x}} \log p(\mathbf{x}, t)$5

where $\nabla_{\mathbf{x}} \log p(\mathbf{x}, t)$6 captures a distribution shift. The shift magnitude is defined as

$\nabla_{\mathbf{x}} \log p(\mathbf{x}, t)$7

Under this setup, the theorem states that, with high probability,

$\nabla_{\mathbf{x}} \log p(\mathbf{x}, t)$8

where $\nabla_{\mathbf{x}} \log p(\mathbf{x}, t)$9 is the number of frames, $-\partial_t \log p(\mathbf{x}, t)$0 is the spatial dimension, $-\partial_t \log p(\mathbf{x}, t)$1 is a lower bound on the stabilized temporal denominator, and $-\partial_t \log p(\mathbf{x}, t)$2 is the failure probability (Zhang et al., 9 Oct 2025).

The significance assigned to this result is that when the generated video is close to the real distribution, with $-\partial_t \log p(\mathbf{x}, t)$3, NSG features are closer, whereas distribution shift enlarges feature distances. The paper interprets this as showing that generated videos exhibit amplified discrepancies in NSG space, thereby justifying the use of MMD on NSG features for detection.

A plausible implication is that the bound functions as a theoretical link between the physical motivation of NSG and the statistical decision rule of NSG-VD: if synthetic generation induces a shift away from natural-video dynamics, then NSG is intended to transform that shift into a more detectable discrepancy.

6. Empirical results, scope, and terminological overlap

The reported experiments state that NSG-VD outperforms state-of-the-art baselines by 16.00% in Recall and 10.75% in F1-Score. The baselines explicitly named are DeMamba, NPR, TALL, and STIL, and the evaluation includes challenging modern generators such as Sora and HotShot. The paper also reports better robustness under data imbalance and stable performance across different thresholds and reference-set sizes (Zhang et al., 9 Oct 2025).

The ablation studies separate spatial gradients alone, temporal derivatives alone, and full NSG. The stated findings are that spatial gradients alone are fairly strong, temporal derivatives alone are weaker, and combining them through NSG gives the best result. This is used to argue that NSG is not merely a reparameterization: the ratio structure that combines spatial and temporal information is what makes it discriminative. Additional ablations indicate that NSG remains effective with BCE, but MMD further improves detection by explicitly comparing the NSG distribution of the test sample against real-video references.

A recurrent source of confusion is terminological overlap. In the 2025 video-detection paper, NSG is a specific probability-flow-based spatiotemporal statistic. By contrast, "Multiobjective Accelerated Gradient-like Flow with Asymptotic Vanishing Normalized Gradient" studies normalized gradient dynamics in multiobjective optimization, with a normalized correction term of the form

$-\partial_t \log p(\mathbf{x}, t)$4

embedded in an inertial continuous-time system; it is related to normalized-gradient lineage, but it is not the same concept as the video-statistic NSG (Yin, 27 Jul 2025). Likewise, "Efficient Sensing of Correlated Spatiotemporal Signals: A Stochastic Gradient Approach" does not use the exact term NSG, although it includes a normalized stochastic-gradient-like adaptation for a contour-margin parameter $-\partial_t \log p(\mathbf{x}, t)$5 in a spatiotemporal sensing system (Alasti, 2019). "Joint direct estimation of 3D geometry and 3D motion using spatio temporal gradients" also does not define NSG; its closest corresponding notion is the normalized spatial gradient direction $-\partial_t \log p(\mathbf{x}, t)$6 used to define normal flow (Barranco et al., 2018).

These distinctions matter. NSG, in the strict sense established by (Zhang et al., 9 Oct 2025), refers to a ratio of spatial probability gradients to temporal density changes for video dynamics. It should not be conflated with normalized gradient corrections in optimization, normalized stochastic-gradient updates in sensing, or normalized image-gradient directions in normal-flow estimation, even though those literatures share related vocabulary.