# Video Temporal Grounding with Multi-Model Collaborative Learning
## 3 Methodology
### Pseudo-Label Generator
#### 3.3.2 Pseudo-Labels Generation  
In the process of pseudo-label generation, we utilize the coarse-grained temporal interval \([t_{cs},t_{ce}]\) provided by the CLIP model and apply normalization to constrain the range of pseudo-label creation, thereby ensuring semantic consistency and producing pseudo-labels \([\hat{t}_{s},\hat{t}_{e}]\) that align with semantic priors. The generation of pseudo-labels depends on the prediction format of different models. Below, we separately describe the pseudo-label generation processes for Span-based models and 2D-MAP models.  
**(1) Pseudo-Labels Generation for Span-based Models**  
As stated in Section 3.2, a Span-based model (such as VSLNet) produces probability distributions \(P_{s}(t)\) and \(P_{e}(t)\) corresponding to the start and end times, respectively. The original probability distributions are normalized over the entire video span \([1,N]\), which contains a substantial number of query-irrelevant time segments. To enhance semantic consistency, it is necessary to constrain and re-normalize these distributions based on the temporal window \([t_{cs},t_{ce}]\) provided by CLIP. Following restriction of these probability distributions to the selected interval, the normalized forms \(\hat{P}_{s}(t)\) and \(\hat{P}_{e}(t)\) are specified as follows:  
\[\hat{P}_{s}(t)=\begin{cases}\dfrac{P_{s}(t)}{\sum_{t=t_{cs}}^{t_{\sigma}}P_{s} (t)},&t\in[t_{cs},t_{\sigma c}],\\ 0,&\text{otherwise}.\end{cases} \tag{10}\]  
\[\hat{P}_{e}(t)=\begin{cases}\dfrac{P_{e}(t)}{\sum_{t=t_{cs}}^{t_{\sigma}}P_{e} (t)},&t\in[t_{cs},t_{\sigma c}],\\ 0,&\text{otherwise}.\end{cases} \tag{11}\]  
The normalization step guarantees that the distributions remain confined to semantically pertinent intervals, thereby narrowing the pseudo-label search domain and fortifying semantic coherence. Upon finalizing the normalization, the pseudo-label boundaries \(\hat{t}_{s}\) and \(\hat{t}_{e}\) emerge by identifying the indices that maximize the normalized probability distributions, formally expressed as follows:  
\[\text{index}_{s}=\arg\max\hat{P}_{s}(t),\quad\text{index}_{e}=\arg\max\hat{P }_{e}(t) \tag{12}\]  
Subsequently, the indices are mapped to pseudo-labels \(PL^{k}\) by Equation (13).  
\[PL^{k}=[\hat{t}_{s}=\text{index}_{s}\times\Delta t,\hat{t}_{e}=\text{index}_{ e}\times\Delta t] \tag{13}\]  
**(2) Pseudo-Labels Generation for 2D-MAP Models**  
Under the 2D-MAP model, the pseudo-labels derive from a 2D temporal probability matrix \(P_{i,j}\), with \(i\) and \(j\) designating the start and end indices, respectively, and adhering to \(i\leq j\). For \(P_{i,j}\), elements within the CLIP-guided temporal window are retained, while probabilities outside this range are set to zero. The remaining elements are then normalized to ensure the sum is one. The definition of the normalized probability matrix \(\hat{P}_{i,j}\) is as follows:  
\[\hat{P}_{i,j}=\begin{cases}\dfrac{P_{i,j}}{\sum_{i=l_{cs}}^{l_{cr}}\sum_{j=i}^{ l_{cr}}\hat{P}_{i,j}},&i,j\in[t_{cs},t_{ce}]\text{ and }i\leq j,\\ 0,&\text{otherwise}.\end{cases} \tag{14}\]  
Subsequently, the highest-probability entry in \(\hat{P}_{i,j}\) pinpoints the start and end indices of the pseudo-labels:  
\[(\text{index}_{i},\text{index}_{j})=\arg\max\hat{P}_{i,j} \tag{15}\]  
Lastly, Equation (16) translates these indices into the pseudo-labels \(PL_{k}\).  
\[PL^{k}=[\hat{t}_{s}=\text{index}_{i}\times\Delta t,\hat{t}_{e}=(\text{index}_ {j}+1)\times\Delta t] \tag{16}\]  
By following the procedure above, we leverage CLIP's semantic guidance to generate high-quality pseudo-labels, thus offering reliable supervisory signals for training.