Complete the definition of Matryoshka representation learning

**Matryoshka representation learning** is a training trick that makes embeddings safe to shorten. A normal embedding is a fixed-size vector and you must use all of it. If you chop off the back half, the remaining numbers no longer form a coherent vector and search quality collapses. A Matryoshka-trained model is different. During training, the loss function looks not just at the full vector but also at several leading prefixes: for example dims 64, 128, 256, 512, 1024, 2048. The full 3072. The model is rewarded for packing the most important information into the front of the vector. By the time training finishes, every prefix is a self-contained embedding you can use directly. The payoff is at deploy time. You can decide how many dimensions you want based on your storage budget, and grab that prefix with a simple slice like `vec[:512]`. No projection matrix, no fine-tuning, no re-embedding the corpus. You pay only the quality cost of using a smaller representation, and that cost is usually small: single-digit recall@10 loss for a 4x truncation.

**Matryoshka** representation learning trains an embedding model so that any leading prefix of the output vector is itself a valid embedding. Instead of a single loss at the full dimension, the training objective sums losses at multiple prefix lengths. Typically a geometric ladder like 64, 128, 256, 512, 1024, 2048, 3072. The model has to pack the most discriminative information into the front of the vector to minimize the joint loss. At deploy time, you pick a dimension that fits your storage or latency budget and truncate with a plain slice. There is no learned projection, no calibration step, no separate fit. The truncated vector lives in the same coordinate system as the full vector, so distances are coherent across truncation levels. The production case is obvious. A 4M-doc corpus at 3072 dims and float32 is 47 GB of raw vectors. The same corpus truncated to 768 dims is 12 GB, which fits comfortably in RAM on a single machine. The quality cost on most well-trained Matryoshka models is 1-3 recall@10 points for a 4x truncation. OpenAI text-embedding-3-large exposes this directly via the `dimensions` parameter; Nomic Embed v1.5 and several open Matryoshka models do the same.