📘

Positional Embedding Interpolation Count

MediumVision-Language Models 📚 VLM 1: Vision-Language Models

Problem Statement

A ViT pretrained at one resolution is fine-tuned at another. Its learned positional embeddings must be interpolated from the old patch grid to the new one. Report how many position vectors exist before and after, keeping any prefix (CLS/register) tokens untouched.

Background

Positional embeddings come as num_prefix + old_grid2** vectors: a few non-spatial prefix tokens (CLS, registers) plus one per patch on a square old_grid x old_grid layout. When the image resolution changes, only the patch positions are bilinearly interpolated to the new_grid x new_grid layout; the prefix embeddings are copied through unchanged. So the new count is num_prefix + new_grid2**.

Your Task

Implement:

def interp_pos_count(total_old, new_grid, num_prefix=1):

total_old: total number of positional embeddings before interpolation.
Infer old_grid from total_old - num_prefix (it is a perfect square).
Return a dict with "old_grid", "new_grid", "total_new" (ints).

Input Format

total_old (int), new_grid (int), num_prefix (int).

Output Format

A dict of three ints.

Sample

print(interp_pos_count(197, 16, 1))

Output:

{'old_grid': 14, 'new_grid': 16, 'total_new': 257}

Example:

Input:

print(interp_pos_count(197, 16, 1))

Output:

{'old_grid': 14, 'new_grid': 16, 'total_new': 257}

Reasoning:

Subtract the number of prefix tokens from the total old embeddings to isolate the spatial patch count: $197 - 1 = 196$ .
Determine the old grid dimension by taking the square root of the patch count, as the patches form a square grid: $\sqrt{196} = 14$ .
Identify the new grid dimension directly from the input parameter: $16$ .
Calculate the total number of new embeddings by adding the unchanged prefix tokens to the new spatial patch count: $1 + 16^2 = 1 + 256 = 257$ .
The final output is {'old_grid': 14, 'new_grid': 16, 'total_new': 257}

Constraints:

total_old - num_prefix is a perfect square (the old grid).
Prefix embeddings pass through; only patch positions change.
total_new = num_prefix + new_grid**2.

More from VLM 1: Vision-Language Models

Visual Token Pruning by Attention MassHard CHAIR-i and CHAIR-s Hallucination RatesMedium Merge a LoRA Update into the Base WeightMedium QLoRA Memory Footprint EstimateMedium POPE Precision, Recall, and F1Medium

1. Background Knowledge

In a Vision Transformer (ViT), an input image is split into non-overlapping square patches. If the image is divided into a grid of size $G \times G$ , there are $G^2$ patches. Each patch is projected into a token, and a positional embedding is added to each token to encode its spatial location. These embeddings are learned parameters during pretraining.

A critical architectural detail is that the positional embedding tensor typically includes prefix tokens at the beginning. The most common is the [CLS] token, which aggregates global image information. Some architectures also include "register" tokens. These prefix tokens are not tied to a specific spatial patch location; they are global. Therefore, the total number of positional embeddings is:

\text{total} = \text{num\_prefix} + G^2

When a ViT pretrained at resolution $G_{\text{old}}$ is fine-tuned at a different resolution $G_{\text{new}}$ , the spatial grid changes. The prefix embeddings (CLS, registers) remain unchanged because they are not spatial. However, the $G_{\text{old}}^2$ spatial embeddings must be interpolated (usually via bilinear interpolation) to fit the new $G_{\text{new}}^2$ grid. This allows the model to transfer knowledge across resolutions without retraining from scratch.

2. Algorithm Approach

This is a straightforward arithmetic inference problem. You are given the total count of embeddings before interpolation and need to reverse-engineer the grid dimensions.

Isolate the spatial component: Subtract the number of prefix tokens from the total to get the number of patch embeddings.
Recover the old grid: Since the patches form a square grid, the number of patches is a perfect square. Take the integer square root to find the old grid size.
Compute the new total: Apply the same formula using the new grid size and the same number of prefix tokens.

No complex interpolation logic is required; you only need to calculate the counts.

👑

Continue the full explanation

You're reading the free preview. Unlock the complete walkthrough, the code editor, test runner and reference solution with Premium.

🔒

Editor locked

The code editor is locked for Pro problems. It is only available for free problems. Please upgrade to gain access to the code editor for all problems.

solution.py

Python 3.13

Test Results

0/0

Run code to see test results.