Matryoshka Representation Learning

Retrieval-augmented generation (RAG) depends on dense retrieval: each document is stored as a learned vector, and a query is answered by finding its nearest neighbors in that vector space. Keeping one full-precision vector per document is the dominant index cost at corpus scale, so retrieval systems replace each vector with a short code of a few bytes---a step called quantization. Standard quantizers such as product quantization (PQ) pick the code that reconstructs the original vector most closely. A single code is even more useful if it serves several byte budgets at once: when its short prefixes are each directly searchable, a deployment can set its efficiency--quality operating point without re-encoding the corpus. But training all prefixes under one objective makes the early bits a compromise across budgets---short codes improve while the full-width code degrades. Quantization to low-bit representation, such as binary codes, further sharpens the conflict. We introduce Matryoshka Hash Representations (MHR), a two-stage procedure that separates full-width training from prefix organization. MHR first learns a longer binary code, then freezes the model and trains additional zero-initialized residual code adaptors for directly searchable prefixes. Documents are stored at one bit per coordinate, while queries keep continuous logits like PQ to attain sufficient expressivity. We implement the search process with FAISS FastScan. Trained on MS MARCO and zero-shot transferred to seven BEIR datasets, MHR reaches .5561 NDCG@10 and .6535 Recall@100 at 32 bytes, surpassing the best baseline of the same budget. The advantage is more pronounced in lower budgets. The same code also strengthens two common pipelines: shortlisting candidates for full-precision reranking, and pruning a low-storage graph index such as LEANN.

Multi-modal late-interaction retrievers achieve strong retrieval on visually rich documents by representing each page as per patch embeddings and matching at the token level. However, this approach incurs high storage costs. Existing compression methods typically fix a single compression level at indexing time, limiting flexibility. We present ColSNAP (Spatial Nested Average Pooling)1, a training method that generates a nested hierarchy of compression levels directly from a backbone's patch grid. By spatially pooling patch embeddings into pro- gressively coarser tiers and training all tiers simultaneously, a single model learns to support retrieval at multiple compression levels without architectural changes. Crucially, a single encoding pass yields every tier, enabling the accuracy-storage trade-off to be configured at indexing time to match avail- able storage budgets, rather than being fixed during training. We demonstrate that models trained using ColSNAP maintain near full-resolution retrieval performance under substantial compression and that ColSNAP transfers effectively across multiple late-interaction backbones, and achieves most of its improvements via a lightweight adaptation stage applied to a pre-trained retriever.

Training a language model suite classically requires training each model separately and serving them independently. We improve both training and inference efficiency by stacking sub-models of increasing size into a single nested architecture trained end-to-end. This Matryoshka training framework reduces the total parameter count of the suite, enables low-cost distillation from the largest to all smaller sub-models at every training step, and is well-suited for speculative decoding as the draft model is contained within the verifier. We validate our approach by training a Matryoshka suite comprising 500M, 1.5B, and 3B sub-models. Our suite is on par with independently trained baselines on benchmark performance and validation and out-of-domain perplexities, while using 36% less training compute and improving the throughput of speculative decoding by 14-26%. We also ablate key architectural choices, offering guidance for building strong Matryoshka LM suites.

A single model scale challenges the flexibility of a production retrieval system: some settings need it faster, others need a smaller index, and the right trade-off changes with the workload. In the context of information retrieval (IR), a transformer-based model can be made smaller in three ways---using fewer layers, passing fewer tokens through the upper layers, or producing a shorter embedding---and each way saves a different compute resource. These options have been studied one at a time, each as its own method with its own code and training setup, which makes them hard to combine or adapt to a new model. We present~\ours to bring all three under one simple abstraction: a single object names any size the model can run at, and a short schedule lists the sizes to train. Training then produces one checkpoint that serves all of those sizes, and at deployment the user picks any of them. The same abstraction covers both retrievers and rerankers and both encoder and decoder models, as it works through interfaces that Hugging Face transformers already expose; a new backbone is a configuration change, not new modeling code. Prior methods---Matryoshka embeddings, early exit, 2DMatryoshka (e.g., Starbucks), and layerwise token compression---become special cases of our unified abstraction. The same interface also enables MatryoshkaLTC (MLTC), which jointly trains several token-compression ratios in one retriever checkpoint. To validate our framework, we train 20 checkpoints across three backbones and two tasks: the quality curves are smooth, one checkpoint costs little over a model trained for a single size, and a controlled study confirms the wallclock speedups. We release the framework and all checkpoints as a resource for building elastic retrieval systems.

Frame selection is essential for applying Large Multimodal Models (LMMs) to long videos due to severe frame redundancy and limited context windows. Since the appropriate frame budget varies with the downstream LMM, reasoning demands, and latency constraints, a practical selector should serve multiple budgets. However, existing methods typically optimize an isolated frame subset for each predefined budget: when the budget changes, previously selected evidence may be replaced rather than progressively augmented. Ranking frames by a fixed score would allow prefix reuse across budgets, but it ignores the distinct roles of different ranking positions. In this paper, we formulate long-video frame selection as a Matryoshka ranking problem: constructing a single priority sequence whose small prefixes concentrate query-conditioned evidence, while progressively larger prefixes preserve this evidence and add broader temporal context. Efficiently constructing such a ranking is itself challenging, as densely sampling long videos and evaluating frame-query relevance incurs substantial overhead. We therefore introduce Matryoshka Evidence-to-Context (MEC) Frame Selection, a training-free framework that builds a reusable sparse video index, discovers candidates through sparse probing and local zooming, and greedily constructs a position-adaptive ranking: early positions emphasize evidence; later positions progressively favor temporal coverage while preserving visual diversity. A single ranking can thus be truncated to any target budget without rerunning the selector. Across four benchmarks and six frame budgets, MEC improves average accuracy over uniform sampling by 3.77 percentage points, matches strong state-of-the-art selectors, and reduces end-to-end selection latency by 47.37-51.19%.

Machine learning engineering (MLE) tasks require long-horizon decision making over iterative solution debugging and refinement, under expensive and feedback-driven environment interactions. Developing and training a monolithic agent for such tasks is fundamentally challenging, as it must simultaneously manage extremely long and noisy contexts, explore vast solution spaces, and remain effective under limited model capacity and computational budgets. To address these challenges, we propose Matryoshka Agent, a unified hierarchical agent framework for complex long-horizon tasks. Matryoshka Agent decomposes agentic problem solving into a coordinated hierarchy of decision making and execution: a high-level Orchestrator maintains compact, long-horizon exploration states and issues strategic instructions, while lower-level Sub-Agents execute concrete solution attempts through direct environment interaction, mediated by standardized Tool interface. This design decouples strategic exploration from costly execution, substantially reducing the burden of long-context reasoning and enabling efficient iterative refinement. We further develop an efficient training paradigm for Matryoshka Agent. Experimental results on a broad range of MLE tasks with diverse model types and scales demonstrate that Matryoshka Agent is an effective and scalable paradigm for long-horizon MLE tasks and complex agentic problem solving. Notably, Matryoshka Agent enables Qwen3-4B-Instruct to reach Orchestrator performance comparable to o4-mini. Applying Matryoshka Agent to Qwen3-30B-Coder results in at most 36.7% relative performance gain.

Metaphor in Arabic is a culturally grounded mechanism for constructing meaning, encoding cultural knowledge that shapes interpretation. Yet current Arabic language models typically collapse lexical, cultural, and metaphorical information into a single representational space, a phenomenon we term "semantic smearing". We introduce CAMMAR (Culture-Aware Matryoshka for Metaphorical Arabic Representations), a representation learning framework that organizes meaning into nested lexical, cultural, and metaphorical embedding subspaces through a staged semantic curriculum. The design implements compositional principles of Al-Jurjani's theory of nazum, modeling figurative meaning as compositionally grounded in prior semantic relations, and yields a training-free geometric measure of metaphoricity based on the distance between lexical and metaphorical representations. Evaluated on a new span-annotated Arabic metaphor set as word-matched figurative/literal pairs, the geometric readout detects metaphor well above chance when the inter-layer geometry is shaped by paired supervision (AUC up to 0.84; figurative outscores its literal counterpart for the same word in 82.6% of pairs), but sits at chance under an unsupervised domain contrast alone, a clean separation between a legible-under-supervision regime and a non-emergent one. A controlled ablation shows that grounding the lexical layer in morphological roots gives a small but consistent gain, an effect absent from direct probing that reflects the layer's quality as a measurement anchor. We will release the datasets, cultural concept inventory, and code upon acceptance.

Differentiable rendering has emerged as a powerful approach for 3D reconstruction and novel view synthesis. State-of-the-art differentiable rendering methods combine a variety of custom representations of 3D geometry and appearance with specialized renderers. However, most downstream tasks in computer graphics rely on 3D meshes. While prior work has attempted differentiable rendering with mesh representations, these approaches are limited to object-centric scenes and fail to reconstruct large-scale, unbounded scenes. In this work, we introduce Meshtryoshka, a novel mesh differentiable rendering framework that combines an off-the-shelf triangle rasterizer with a 3D representation that consists of nested mesh shells which resemble a matryoshka doll. In every forward pass, the mesh shells are extracted anew from a 3D signed distance function via iso-surface extraction, and the opacities for each vertex are computed as a function of signed distance. Each mesh shell is then rasterized independently, and the final image is created via alpha compositing. Crucially, mesh vertex positions are updated only indirectly via gradients that flow through the opacity values into the signed distance function, and hence, our method is compatible with off-the-shelf mesh renderers that need not be differentiable with respect to vertex positions. On object-centric scenes, our method performs competitively with surface-based differentiable rendering techniques. Our differentiable mesh rendering method scales to unbounded, real-world 3D scenes, where it yields high-quality novel view synthesis results approaching those of state-of-the-art, non-mesh methods. Our method suggests that it may be possible to solve the differentiable rendering problem without relying on specialized renderers, only using conventional tools from the computer graphics toolbox.

Multi-vector visual document retrievers achieve strong fine-grained matching by representing each page with multiple vectors from deep Vision-Language Models (VLMs), but this design makes deployment expensive in both storage and computational overhead. Existing efficiency techniques usually optimize only part of this budget, leaving multimodal retrievers without a unified way to trade accuracy for both vector width and encoder depth. Therefore, we propose MM-Matryoshka, a 2D Matryoshka training framework for budget-elastic Visual Document Retrieval (VDR), enabling ColPali-style multi-vector retrieval elastic along both dimension and layer. At inference time, a single retriever can select a 2D selectable budget without training separate models for different budgets. Through comprehensive experiments across multiple representative backbones, we demonstrate that by retaining significantly higher quality than direct truncation baselines while substantially reducing storage and computational overhead, MM-Matryoshka can offer robust budget elasticity for efficient VDR.

Concept Bottleneck Models (CBMs) have emerged as a prominent paradigm for interpretable deep learning, learning by grounding predictions in human-understandable concepts. However, their practical deployment is hindered by the high cost of test-time intervention, as correcting model errors typically requires human experts to manually inspect and verify a large set of predicted concepts. Existing approaches suffer from a fundamental structural limitation: they either adopt a single static concept set, forcing experts to exhaustively annotate concepts and incurring prohibitive intervention costs, or train multiple models tailored to different concept budgets, resulting in substantial computational and maintenance overhead. To address this challenge, we propose the Matryoshka Concept Bottleneck Model (MCBM), a unified architecture that enables adaptive concept utilization within a single model. Inspired by Matryoshka Representation Learning, MCBM organizes concepts into a nested hierarchy based on maximum relevance and minimum redundancy, allowing inference at multiple levels of conceptual granularity without retraining. Theoretically, we show that MCBM reduces the expected intervention costs from linear to logarithmic order, $O(\log K)$, while guaranteeing monotonic performance improvement. Empirically, extensive experiments demonstrate that MCBM matches the performance of independently trained models while enabling dynamic and efficient expert interaction.

Embedding models are pivotal in industrial information retrieval systems like search and advertising. However, existing pretrained models often exhibit fixed architectures and embedding dimensionalities, posing significant challenges when adapting them to diverse deployment scenarios with varying business-driven constraints. A common practice involves fine-tuning with partial parameter initialization from larger pretrained models for resource-constrained tasks. This method is often suboptimal as the misalignment between pretraining and downstream usage prevents full realization of pretraining benefits. To address this limitation, we introduce m3BERT: a Modern, Multi-lingual, Matryoshka Bidirectional Encoder, which features a novel pretraining strategy that jointly optimizes representations across both transformer layers and multiple embedding dimensions. This enables a single model to be tailored to varied resource and accuracy targets while maintaining consistency with pretraining. Incorporating recent architectural improvements, m3BERT uses a three-stage pretraining: monolingual pretraining, multilingual adaptation to serve diverse user bases, and crucial continual pretraining on a massive web domain corpus to enhance utility in commercial retrieval. m3BERT significantly outperforms state-of-the-art embedding models in Bing-Click, a large-scale industrial retrieval dataset, showcasing its practical versatility as an efficient foundation for resource-aware industrial retrieval systems. Further experiments on public datasets also confirm the general effectiveness of our multigranular Matryoshka pretraining strategy.

Frozen vision-language embeddings contain signals at multiple semantic resolutions, from object identity to attributes, relations, and full-caption meaning, but they expose these signals through a fixed-length vector interface. We study whether embedding length can be turned into a controllable semantic access interface. We propose \textbf{GraSP-VL}, which learns a shared near-orthogonal prefix transform over frozen VLM embeddings. GraSP-VL instantiates a \textbf{Semantic Matryoshka} interface: short prefixes are assigned coarse semantic roles, while longer prefixes progressively expose finer language-grounded distinctions. Because the transform is shared across image and text embeddings and preserves full-dimensional geometry, prefix behavior changes without rewriting the original VLM space. On a 20,147-example COCO/Flickr30K annotation pool, GraSP-VL reaches a staircase score of 53.01 and hard-negative selectivity of 89.76, while keeping full-space drift below $10^{-6}$. It also transfers to SugarCrepe-clean with 86.03 object accuracy and 11.96 mean external emergence, and preserves full-dimensional zero-shot CIFAR-100 accuracy. These results show that frozen VLM embeddings can be reorganized into a truncatable semantic prefix interface rather than merely compressed.

Matryoshka Representation Learning (MRL) is a widely adopted approach for training text encoders so they provide useful text representations at various sizes, available by simply truncating the resulting vectors at sizes pre-determined at training time. Recent works have shown that randomly truncating text embeddings has minimal impact in downstream performance unless vectors are reduced in size by at least 70%, suggesting that embeddings are already robust to truncation without the use of MRL. However, no prior work has compared random truncation to MRL, so it is unclear how the two methods compare as effective embedding reduction methods. In this paper, we study this by applying the same truncation used by MRL to models trained with and without MRL. Our results across several models and downstream tasks show that, unless heavily truncating embeddings (i.e. reducing their size by at least 80%), truncated embeddings of non-MRL models are competitive with, and often outperform models trained with MRL. This suggests that truncation robustness may not necessarily come from MRL, and that the choice of spending the additional training cost of MRL depends on whether heavy truncation is desired. We make our code available for reproduction.

The development of high-quality text embeddings is increasingly drifting toward an exclusionary future, defined by three critical barriers: prohibitive computational costs, a narrow linguistic focus that neglects most of the world's languages, and a lack of transparency from closed-source or open-weight models that stifles research. To dismantle these barriers, we introduce ML-Embed, a suite of inclusive and efficient models built upon a new framework: 3-Dimensional Matryoshka Learning (3D-ML). Our framework addresses the computational challenge with comprehensive efficiency across the entire model lifecycle. Beyond the storage benefits of Matryoshka Representation Learning (MRL) and flexible inference-time depth provided by Matryoshka Layer Learning (MLL), we introduce Matryoshka Embedding Learning (MEL) for enhanced parameter efficiency. To address the linguistic challenge, we curate a massively multilingual dataset and train a suite of models ranging from 140M to 8B parameters. In a direct commitment to transparency, we release all models, data, and code. Extensive evaluation on 430 tasks demonstrates that our models set new records on 9 of 17 evaluated MTEB benchmarks, with particularly strong results in low-resource languages, providing a reproducible blueprint for building globally equitable and computationally efficient AI systems.

Learned representations are often invariant to rotational transformations, leaving individual dimensions non-identifiable and interchangeable. We study how Matryoshka Representation Learning (MRL) induces a task-aligned privileged basis distinct from variance-based or regularizer-induced orderings. In the linear setting, we prove that full-prefix MRL recovers the ordered principal directions, and can be computed efficiently using shared statistics. Empirically, we demonstrate that MRL yields consistent per-dimension structure aligned with task signal, where coordinate magnitude reflects informativeness.

With the rise in scale for deep learning models to billions of parameters, the computational cost of fine-tuning remains a significant barrier to deployment. While Low-Rank Adaptation (LoRA) has become the standard for parameter-efficient fine-tuning, the need to set a predefined, static rank $r$ requires exhaustive grid searches to balance efficiency and performance. Existing rank-adaptive solutions such as DyLoRA mitigate this by sampling ranks during the training from a predefined distribution. However, they often yield sub-optimal results at higher ranks due to lack of consistent gradient signals across the full hierarchy of ranks, thus making these methods data-inefficient. In this paper, we propose MatryoshkaLoRA, a general, Matryoshka-inspired training framework for LoRA that learns accurate hierarchical low-rank representations by inserting a fixed, carefully crafted diagonal matrix $P$ between the existing LoRA adapters to scale their sub-ranks accordingly. By introducing this simple modification, our general framework recovers LoRA and DyLoRA only by changing $P$ and ensures all sub-ranks embed the available gradient information efficiently. Our MatryoshkaLoRA supports dynamic rank selection with minimal degradation in accuracy. We further propose Area Under the Rank Accuracy Curve (AURAC), a metric that consistently evaluates the performance of hierarchical low-rank adapters. Our results demonstrate that MatryoshkaLoRA learns more accurate hierarchical low-rank representations than prior rank-adaptive approaches and achieves superior accuracy-performance trade-offs across ranks on the evaluated datasets. Our code is available at https://github.com/IST-DASLab/MatryoshkaLoRA.

Representation learning is fundamental to NLP, but building embeddings that work well at different computational budgets is challenging. Matryoshka Representation Learning (MRL) offers a flexible inference paradigm through nested embeddings; however, learning such structures requires explicit coordination of how information is arranged across embedding dimensionality and model depth. In this work, we propose MIPIC (Matryoshka Representation Learning via Self-Distilled Intra-Relational Alignment and Progressive Information Chaining), a unified training framework designed to produce structurally coherent and semantically compact Matryoshka representations. MIPIC promotes cross-dimensional structural consistency through Self-Distilled Intra-Relational Alignment (SIA), which aligns token-level geometric and attention-driven relations between full and truncated representations using top-k CKA self-distillation. Complementarily, it enables depth-wise semantic consolidation via Progressive Information Chaining (PIC), a scaffolded alignment strategy that incrementally transfers mature task semantics from deeper layers into earlier layers. Extensive experiments on STS, NLI, and classification benchmarks (spanning models from TinyBERT to BGEM3, Qwen3) demonstrate that MIPIC yields Matryoshka representations that are highly competitive across all capacities, with significant performance advantages observed under extreme low-dimensional.