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+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0000", "section": "ABSTRACT", "page_start": 1, "page_end": 1, "type": "Text", "text": "The transformer architecture has demonstrated remarkable capabilities in modern artificial intelligence, among which the capability of implicitly learning an internal model during inference time is widely believed to play a key role in the understanding of pre-trained large language models. However, most recent works have been focusing on studying supervised learning topics such as in-context learning, leaving the field of unsupervised learning largely unexplored. This paper investigates the capabilities of transformers in solving Gaussian Mixture Models (GMMs), a fundamental unsupervised learning problem through the lens of statistical estimation. We propose a transformer-based learning framework called Transformer for Gaussian Mixture Models (TGMM) that simultaneously learns to solve multiple GMM tasks using a shared transformer backbone. The learned models are empirically demonstrated to effectively mitigate the limitations of classical methods such as Expectation-Maximization (EM) or spectral algorithms, at the same time exhibit reasonable robustness to distribution shifts. Theoretically, we prove that transformers can efficiently approximate both the Expectation-Maximization (EM) algorithm and a core component of spectral methods—namely, cubic tensor power iterations. These results not only improve upon prior work on approximating the EM algorithm, but also provide, to our knowledge, the first theoretical guarantee that transformers can approximate high-order tensor operations. Our study bridges the gap between practical success and theoretical understanding, positioning transformers as versatile tools for unsupervised learning. 1", "source": "marker_v2", "marker_block_id": "/page/0/Text/7"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0001", "section": "1 INTRODUCTION", "page_start": 1, "page_end": 1, "type": "Text", "text": "Large Language Models (LLMs) have achieved remarkable success across various tasks in recent years. Transformers (Vaswani et al., 2017) , the dominant architecture in modern LLMs (Brown et al., 2020) , outperform many other neural network models in efficiency and scalability. Beyond language tasks, transformers have also demonstrated strong performance in other domains, such as computer vision (Han et al., 2023; Khan et al., 2022) and reinforcement learning (Li et al., 2023a) . Given their practical success, understanding the mechanisms behind transformers has attracted growing research interest. Existing studies often treat transformers as algorithmic toolboxes, investigating their ability to implement diverse algorithms (Von Oswald et al., 2023; Bai et al., 2023; Lin et al., 2024; Giannou et al., 2025; Teh et al., 2025) –a perspective linked to meta-learning (Hospedales et al., 2021) .", "source": "marker_v2", "marker_block_id": "/page/0/Text/9"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0002", "section": "1 INTRODUCTION", "page_start": 1, "page_end": 1, "type": "Text", "text": "However, most research has focused on supervised learning settings, such as regression (Bai et al., 2023) and classification (Giannou et al., 2025) , leaving the unsupervised learning paradigm relatively unexplored. Since transformer models are typically trained in a supervised manner, unsupervised learning poses inherent challenges for transformers due to the absence of labeled data. Moreover, given the abundance of unlabeled data in real-world scenarios, investigating the mechanisms of", "source": "marker_v2", "marker_block_id": "/page/0/Text/10"}
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+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0004", "section": "1 INTRODUCTION", "page_start": 1, "page_end": 1, "type": "Footnote", "text": "1 Code available at", "source": "marker_v2", "marker_block_id": "/page/0/Footnote/13"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0005", "section": "1 INTRODUCTION", "page_start": 2, "page_end": 2, "type": "Text", "text": "transformers in unsupervised learning holds significant implications for practical applications. The Gaussian mixture model (GMM) represents one of the most fundamental unsupervised learning tasks in statistics, with a rich historical background (DAY, 1969; Aitkin & Wilson, 1980) and ongoing research interest (Zhang et al., 2021; Manduchi et al., 2021; Löffler et al., 2021; Ndaoud, 2022; Gribonval et al., 2021; Yu et al., 2021) . Two primary algorithmic approaches are existing for solving GMM problems: (1) likelihood-based methods employing the Expectation-Maximization (EM) algorithm (Dempster et al., 1977; Balakrishnan et al., 2017) , and (2) moment-based methods utilizing spectral algorithms (Hsu & Kakade, 2013; Anandkumar et al., 2014) . However, both algorithms have inherent limitations. The EM algorithm is prone to convergence at local optima and is highly sensitive to initialization (Moitra, 2018; Jin et al., 2016) . In contrast, while the spectral method is independent of initialization, it requires the number of components to be smaller than the data's dimensionality—an assumption that restricts its applicability to problems involving many components in low-dimensional GMMs (Hsu & Kakade, 2013) .", "source": "marker_v2", "marker_block_id": "/page/1/Text/1"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0006", "section": "1 INTRODUCTION", "page_start": 2, "page_end": 2, "type": "Text", "text": "In this work, we explore transformers for GMM parameter estimation to address two questions. (i) Can Transformers provably work for GMM in-context? (ii) Can Transformers empirically overcome the drawbacks of both EM algorithm and the spectral method? Our answers are affirmative. We find that meta-trained transformers exhibit strong performance on GMM tasks without the aforementioned limitations. Notably, we construct transformer-based solvers that efficiently solve GMMs with varying component counts simultaneously. The experimental phenomena are further backed up by novel theoretical establishments: We prove that transformers can effectively learn GMMs with different components by approximating both the EM algorithm and a key component of spectral methods on GMM tasks.", "source": "marker_v2", "marker_block_id": "/page/1/Text/2"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0007", "section": "Main Contributions.", "page_start": 2, "page_end": 2, "type": "ListGroup", "text": "We propose the TGMM framework that utilizes transformers to solve multiple GMM tasks with varying numbers of components simultaneously during inference time. Through extensive experimentation, the learned TGMM model is demonstrated to achieve competitive and robust performance over synthetic GMM tasks. Notably, TGMM outperforms the popular EM algorithm in terms of estimation quality, and approximately matches the strong performance of spectral methods while enjoying better flexibility. We establish theoretical foundations by proving that transformers can approximate both the EM algorithm and a key component of spectral methods. Our approximation of the EM algorithm fundamentally leverages the weighted averaging property inherent in softmax attention, enabling simultaneous approximation of both the E and M steps. Notably, our approximation results also hold across varying dimensions and mixture components in GMM. We proved that transformers (with RELU activation) can implement cubic tensor power iterationsa crucial component of spectral algorithms for GMM. The proof is highly dependent on the multi-head structure of transformers. To the best of our knowledge, this is the first theoretical demonstration of transformers' capacity for high-order tensor calculations.", "source": "marker_v2", "marker_block_id": "/page/1/ListGroup/316"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0008", "section": "Main Contributions.", "page_start": 2, "page_end": 2, "type": "Text", "text": "Related works. Recent research has explored the mechanisms by which transformers can implement various supervised learning algorithms. For instance, Akyürek et al. (2023) , Von Oswald et al. (2023) , and Bai et al. (2023) demonstrate that transformers can perform gradient descent for linear regression problems in-context. Lin et al. (2024) shows that transformers are capable of implementing Upper Confidence Bound (UCB) algorithms, as well as other classical algorithms in reinforcement learning tasks. Giannou et al. (2025) reveals that transformers can execute in-context Newton's method for logistic regression problems. Teh et al. (2025) illustrates that transformers can approximate Robbins' estimator and solve Naive Bayes problems. Kim et al. (2024) studies the minimax optimality of transformers on nonparametric regression. Some literature on density estimation using LLMs is discussed in Section A.", "source": "marker_v2", "marker_block_id": "/page/1/Text/7"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0009", "section": "Main Contributions.", "page_start": 2, "page_end": 2, "type": "Text", "text": "Comparison with prior theoretical works in unsupervised learning setting. Several recent studies have investigated the mechanisms of transformer-based models in mixture model settings (He et al., 2025a; Jin et al., 2024; He et al., 2025b) . Among these, He et al. (2025a) establishes that transformers can implement Principal Component Analysis (PCA) and leverages this to GMM clustering. However, their analysis is limited to the two-component case, restricting its broader applicability.", "source": "marker_v2", "marker_block_id": "/page/1/Text/8"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0010", "section": "Main Contributions.", "page_start": 2, "page_end": 2, "type": "Text", "text": "The paper Jin et al. (2024) investigates the in-context learning capabilities of transformers for mixture linear models, a setting that differs from ours. Furthermore, their approximation construction of the transformer is limited to two-component GMMs, leaving the general case unaddressed. While", "source": "marker_v2", "marker_block_id": "/page/1/Text/9"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0011", "section": "Main Contributions.", "page_start": 3, "page_end": 3, "type": "Text", "text": "they assume ReLU as the activation function—contrary to the conventional choice of softmax—their theoretical proofs rely on a key lemma from prior work Pathak et al. (2024) that assumes softmax activation, thereby introducing an inconsistency in their assumptions. The paper He et al. (2025b) studies the performance of transformers on multi-class GMM clustering, a setting closely related to ours. However, our work focuses on parameter estimation rather than clustering . We give a discussion of our theoretical improvements over their work in detail in the following paragraph. From an empirical perspective, their experiments are conducted on a small-scale transformer, which fails to validate their theoretical claims.", "source": "marker_v2", "marker_block_id": "/page/2/Text/1"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0012", "section": "Main Contributions.", "page_start": 3, "page_end": 3, "type": "Text", "text": "Sharpness of our results. Our theoretical analysis fully leverages key architectural components of Transformers: the query-key-value mechanism, multi-head attention, and the properties of the activation function. It is worth pointing out that our result improves the prior work for EM approximation in several points: First, Our analysis shows that Transformers can approximate L-step EM algorithms with just O(L) layers, a significant improvement over prior work (He et al., 2025b), which requires O(KL) layers (dependent on the number of components K). Second, unlike He et al. (2025b), which needs number of attention heads M \\to +\\infty to get valid bounds, our results hold with M = O(1), aligning better with real-world designs. Third, our approximation bounds scale polynomially in dimension d, unlike He et al. (2025b)'s exponential dependence—a crucial improvement for high-dimensional settings. We believe our results and proofs can offer profound insights for subsequent theoretical research on transformers.", "source": "marker_v2", "marker_block_id": "/page/2/Text/2"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0013", "section": "Main Contributions.", "page_start": 3, "page_end": 3, "type": "Text", "text": "Organization . The rest of paper is organized as follows. In Section 2, some background knowledge is introduced. In Section 3, we present the experimental details and findings. The theoretical results are proposed in Section 4, and some discussions are given in Section 5. The proofs and additional experimental results are given in the appendix.", "source": "marker_v2", "marker_block_id": "/page/2/Text/3"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0014", "section": "Main Contributions.", "page_start": 3, "page_end": 3, "type": "Text", "text": "Notations. We introduce the following notations. Let [n] := \\{1, 2, \\dots, n\\} . All vectors are represented as column vectors unless otherwise specified. For a vector v \\in \\mathbb{R}^d , we denote ||v|| as its Euclidean norm. For two sequences a_n and b_n indexed by n, we denote a_n = O(b_n) if there exists a universal constant C such that a_n \\leq Cb_n for sufficiently large n.", "source": "marker_v2", "marker_block_id": "/page/2/Text/4"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0015", "section": "2.1 PRELIMINARIES", "page_start": 3, "page_end": 3, "type": "Text", "text": "The Gaussian mixture model (GMM) is a cornerstone of unsupervised learning in statistics, with deep historical roots and enduring relevance in modern research. Since its early formalizations (DAY, 1969; Aitkin & Wilson, 1980), GMM has remained a fundamental tool for clustering and density estimation, widely applied across diverse domains. Recent advances have further explored the theoretical foundations of Gaussian Mixture Models (GMMs)(Löffler et al., 2021; Ndaoud, 2022; Gribonval et al., 2021), extended their applications in incomplete data settings (Zhang et al., 2021), and integrated them with deep learning frameworks (Manduchi et al., 2021; Yu et al., 2021). Due to their versatility and interpretability, GMMs remain indispensable in unsupervised learning, effectively bridging classical statistical principles with modern machine learning paradigms. We consider the (unit-variance) isotropic Gaussian Mixture Model with K components, with its probability density function as", "source": "marker_v2", "marker_block_id": "/page/2/Text/7"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0016", "section": "2.1 PRELIMINARIES", "page_start": 3, "page_end": 3, "type": "Equation", "text": "p(x|\\boldsymbol{\\theta}) = \\sum_{k=1}^{K} \\pi_k \\phi(x; \\mu_k) , \\qquad (1)", "source": "marker_v2", "marker_block_id": "/page/2/Equation/8"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0017", "section": "2.1 PRELIMINARIES", "page_start": 3, "page_end": 3, "type": "Text", "text": "where \\phi(x;\\mu) is the standard Gaussian kernel, i.e. \\phi(x;\\mu) = \\frac{1}{(2\\pi)^{d/2}} \\exp\\left(-\\frac{1}{2}(x-\\mu)^{\\top}(x-\\mu)\\right) . The parameter \\boldsymbol{\\theta} is defined as \\boldsymbol{\\theta} = \\boldsymbol{\\pi} \\cup \\boldsymbol{\\mu} , where \\boldsymbol{\\pi} := \\{\\pi_1, \\pi_2, \\cdots, \\pi_K\\}, \\, \\pi_k \\in \\mathbb{R} and \\boldsymbol{\\mu} = \\{\\mu_1, \\mu_2, \\cdots, \\mu_K\\}, \\mu_k \\in \\mathbb{R}^d, \\, k \\in [K] . We take N samples \\mathbf{X} = \\{X_i\\}_{i \\in [N]} from model (1). \\{X_i\\}_{i \\in [N]} can be also rewritten as", "source": "marker_v2", "marker_block_id": "/page/2/Text/9"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0018", "section": "2.1 PRELIMINARIES", "page_start": 3, "page_end": 3, "type": "Equation", "text": "X_i = \\mu_{y_i} + Z_i,", "source": "marker_v2", "marker_block_id": "/page/2/Equation/10"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0019", "section": "2.1 PRELIMINARIES", "page_start": 3, "page_end": 3, "type": "Text", "text": "where \\{y_i\\}_{i\\in[N]} are i.i.d. discrete random variables with \\mathbb{P}(y=k)=\\pi_k for k\\in[K] and \\{Z_i\\}_{i\\in[N]} are i.i.d. standard Gaussian random vector in \\mathbb{R}^d .", "source": "marker_v2", "marker_block_id": "/page/2/Text/11"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0020", "section": "2.1 PRELIMINARIES", "page_start": 3, "page_end": 3, "type": "Text", "text": "The EM algorithm(Dempster et al., 1977) remains the most widely used approach for GMM parameter estimation. Due to space constraints, we propose the algorithm in Section B. Alternatively, the", "source": "marker_v2", "marker_block_id": "/page/2/Text/12"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0021", "section": "2.1 PRELIMINARIES", "page_start": 4, "page_end": 4, "type": "Text", "text": "spectral algorithm(Hsu & Kakade, 2013) offers an efficient moment-based approach that estimates parameters through low-order observable moments. A key component of this method is cubic tensor decomposition(Anandkumar et al., 2014). For brevity, we defer the algorithmic details to Section B.", "source": "marker_v2", "marker_block_id": "/page/3/Text/1"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0022", "section": "2.1 PRELIMINARIES", "page_start": 4, "page_end": 4, "type": "Text", "text": "Next, we give a rigorous definition of the transformer model. To maintain consistency with existing literature, we adopt the notational conventions presented in Bai et al. (2023), with modifications tailored to our specific context. We consider a sequence of N input vectors \\{h_i\\}_{i=1}^N \\subset \\mathbb{R}^D , which can be compactly represented as an input matrix \\mathbf{H} = [h_1, \\dots, h_N] \\in \\mathbb{R}^{D \\times N} , where each h_i corresponds to a column of \\mathbf{H} (also referred to as a token).", "source": "marker_v2", "marker_block_id": "/page/3/Text/2"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0023", "section": "2.1 PRELIMINARIES", "page_start": 4, "page_end": 4, "type": "Text", "text": "Here we introduce several useful definitions and their full notations are given in Appendix C.", "source": "marker_v2", "marker_block_id": "/page/3/Text/3"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0024", "section": "2.1 PRELIMINARIES", "page_start": 4, "page_end": 4, "type": "Text", "text": "Definition 1 (Attention layer). A (self-)attention layer with M heads is denoted as \\operatorname{Attn}_{\\Theta_{\\operatorname{attn}}}(\\cdot) with parameters \\Theta_{\\operatorname{attn}} = \\{(\\mathbf{V}_m, \\mathbf{Q}_m, \\mathbf{K}_m)\\}_{m \\in [M]} \\subset \\mathbb{R}^{D \\times D} .", "source": "marker_v2", "marker_block_id": "/page/3/Text/4"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0025", "section": "2.1 PRELIMINARIES", "page_start": 4, "page_end": 4, "type": "Text", "text": "Definition 2 (MLP layer). A (token-wise) MLP layer with hidden dimension D' is denoted as \\mathrm{MLP}_{\\mathbf{\\Theta}_{\\mathtt{nlp}}}(\\cdot) with parameters \\mathbf{\\Theta}_{\\mathtt{nlp}} = (\\mathbf{W}_1, \\mathbf{W}_2) \\in \\mathbb{R}^{D' \\times D} \\times \\mathbb{R}^{D \\times D'} .", "source": "marker_v2", "marker_block_id": "/page/3/Text/5"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0026", "section": "2.1 PRELIMINARIES", "page_start": 4, "page_end": 4, "type": "Text", "text": "Definition 3 (Transformer). An L-layer transformer, denoted as \\mathrm{TF}_{\\Theta_{\\mathrm{TF}}}(\\cdot) , is a composition of L self-attention layers each followed by an MLP layer:", "source": "marker_v2", "marker_block_id": "/page/3/Text/6"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0027", "section": "2.1 PRELIMINARIES", "page_start": 4, "page_end": 4, "type": "Equation", "text": "\\mathrm{TF}_{\\boldsymbol{\\Theta}_{\\mathrm{TF}}}(\\mathbf{H}) = \\mathrm{MLP}_{\\boldsymbol{\\Theta}_{\\mathrm{nlp}}^{(L)}} \\Big( \\mathrm{Attn}_{\\boldsymbol{\\Theta}_{\\mathrm{attn}}^{(L)}} \\Big( \\cdots \\mathrm{MLP}_{\\boldsymbol{\\Theta}_{\\mathrm{nlp}}^{(1)}} \\Big( \\mathrm{Attn}_{\\boldsymbol{\\Theta}_{\\mathrm{attn}}^{(1)}} (\\mathbf{H}) \\Big) \\Big) \\Big).", "source": "marker_v2", "marker_block_id": "/page/3/Equation/7"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0028", "section": "2.2 THE TGMM ARCHITECTURE", "page_start": 4, "page_end": 4, "type": "Text", "text": "A recent line of work(Xie et al., 2021; Garg et al., 2022; Bai et al., 2023; Akyürek et al., 2023; Li et al., 2023b) has been studying the capability of transformer that functions as a data-driven algorithm under the context of in-context learning (ICL). However, in contrast to the setups therein where inputs consist of both features and labels, under the unsupervised GMM setup, there is no explicitly provided label information. Therefore, we formulate the learning problem as learning an estimation algorithm instead of learning a prediction algorithm as in the case of ICL. A notable property of GMM is that the structure of the estimand depends on an unknown parameter K, which is often treated as a hyper-parameter in GMM estimation(Titterington et al., 1985; McLachlan & Peel, 2000). For clarity of representation, we define an isotropic Gaussian mixture task as \\mathcal{T} = (\\theta, \\mathbf{X}, K) , where \\mathbf{X} is a i.i.d. sample generated according to ground truth \\theta according to the isotropic GMM law and K is the configuration used during estimation which we assume to be the same as the number of components of the ground truth \\theta . The GMM task is solved via applying some algorithm \\mathcal{A} that takes \\mathbf{X} and K as inputs and outputs an estimate of the ground truth \\hat{\\theta} = \\mathcal{A}(\\mathbf{X}; K) .", "source": "marker_v2", "marker_block_id": "/page/3/Text/9"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0029", "section": "2.2 THE TGMM ARCHITECTURE", "page_start": 4, "page_end": 4, "type": "Text", "text": "In this paper, we propose a transformer-based architecture, transformers-for-Gaussian-mixtures (TGMM), as a GMM task solver that allows flexibility in its outputs, while at the same time being parameter-efficient, as illustrated in Figure 1: A TGMM model supports solving s different GMM tasks with K \\in \\mathcal{K} := \\{K_1, \\ldots, K_s\\} . Given inputs N data points \\mathbf{X} \\in \\mathbb{R}^{d \\times N} and a structure configuration of the estimand K. TGMM first augments the inputs with auxiliary configurations about K via concatenating it with a task embedding \\mathbf{P} = \\text{embed}(K) , i.e., \\mathbf{H} = [\\mathbf{X}||\\mathbf{P}] , and use a linear Readin layer to project the augmented inputs onto a shared hidden representation space for several estimand structures \\{K_1, \\ldots, K_s\\} , which is then manipulated by a shared transformer backbone that produces task-aware hidden representations. The TGMM estimates are then decoded by task-specific Readout modules. More precisely, with target decoding parameters of K components, the Readout module first performs an attentive-pooling operation (Lee et al., 2019):", "source": "marker_v2", "marker_block_id": "/page/3/Text/10"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0030", "section": "2.2 THE TGMM ARCHITECTURE", "page_start": 4, "page_end": 4, "type": "Equation", "text": "\\mathbf{O} = (\\mathbf{V}_o \\mathbf{H}) \\mathrm{SoftMax} ((\\mathbf{K}_o \\mathbf{H})^{\\top} \\mathbf{Q}_o) \\in \\mathbb{R}^{(d+K) \\times K},", "source": "marker_v2", "marker_block_id": "/page/3/Equation/11"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0031", "section": "2.2 THE TGMM ARCHITECTURE", "page_start": 4, "page_end": 4, "type": "Text", "text": "where \\mathbf{V}_o, \\mathbf{K}_o \\in \\mathbb{R}^{(d+K)\\times D} , \\mathbf{Q}_o \\in \\mathbb{R}^{(d+K)\\times K} . The estimates for mixture probability are then extracted by a row-wise mean-pooling of the first K rows of \\mathbf{O} , and the estimates for mean vectors are the last d rows of \\mathbf{O} . We wrap the above procedure as \\{\\widehat{\\pi}_k, \\widehat{\\mu}_k\\}_{i\\in[K]} = \\operatorname{Readout}_{\\Theta_{\\text{out}}}(\\mathbf{H}) . TGMM is parameter-efficient in the sense that it only introduces extra parameter complexities of the order O(sdD) in addition to the backbone. We give a more detailed explanation of the parameter efficiency of TGMM in appendix Section D. We wrap the TGMM model into the following form:", "source": "marker_v2", "marker_block_id": "/page/3/Text/12"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0032", "section": "2.2 THE TGMM ARCHITECTURE", "page_start": 4, "page_end": 4, "type": "Equation", "text": "\\mathrm{TGMM}_{\\mathbf{\\Theta}}(\\mathbf{X}; K) = \\mathrm{Readout}_{\\mathbf{\\Theta}_{out}}(\\mathrm{TF}_{\\mathbf{\\Theta}_{TF}}(\\mathrm{Readin}_{\\mathbf{\\Theta}_{in}}([\\mathbf{X}||\\mathrm{embed}(K)]))).", "source": "marker_v2", "marker_block_id": "/page/3/Equation/13"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0033", "section": "2.2 THE TGMM ARCHITECTURE", "page_start": 4, "page_end": 4, "type": "Text", "text": "Above, the parameter \\Theta = (\\Theta_{TF}, \\Theta_{in}, \\Theta_{out}) consists of the parameters in the transformer \\Theta_{TF} and the parameters in the Readin and the Readout functions \\Theta_{in}, \\Theta_{out} .", "source": "marker_v2", "marker_block_id": "/page/3/Text/14"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0034", "section": "2.2 THE TGMM ARCHITECTURE", "page_start": 5, "page_end": 5, "type": "FigureGroup", "text": "Figure 1: Illustration of the proposed TGMM architecture: TGMM utilizes a shared transformer backbone that supports solving s different kind of GMM tasks via a task-specific Readout strategies.", "source": "marker_v2", "marker_block_id": "/page/4/FigureGroup/247"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0035", "section": "Algorithm 1 TaskSampler", "page_start": 5, "page_end": 5, "type": "Text", "text": "Require: sampling", "source": "marker_v2", "marker_block_id": "/page/4/Text/4"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0036", "section": "Algorithm 1 TaskSampler", "page_start": 5, "page_end": 5, "type": "Text", "text": "distributions", "source": "marker_v2", "marker_block_id": "/page/4/Text/5"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0037", "section": "Algorithm 1 TaskSampler", "page_start": 5, "page_end": 5, "type": "Text", "text": "p_{\\mu}, p_{\\pi}, p_N, p_K .", "source": "marker_v2", "marker_block_id": "/page/4/Text/6"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0038", "section": "Algorithm 1 TaskSampler", "page_start": 5, "page_end": 5, "type": "ListGroup", "text": "1: Sample the type of task (i.e., number of mixture components) K \\sim p_K . Sample a GMM task according to the type of task", "source": "marker_v2", "marker_block_id": "/page/4/ListGroup/248"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0039", "section": "Algorithm 1 TaskSampler", "page_start": 5, "page_end": 5, "type": "Equation", "text": "\\bm{\\theta} = (\\bm{\\mu}, \\bm{\\pi}), \\bm{\\mu} \\sim p_{\\mu}, \\bm{\\pi} \\sim p_{\\pi},", "source": "marker_v2", "marker_block_id": "/page/4/Equation/9"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0040", "section": "Algorithm 1 TaskSampler", "page_start": 5, "page_end": 5, "type": "Text", "text": "where \\boldsymbol{\\mu} = \\{\\mu_1, \\cdots, \\mu_K\\}, \\boldsymbol{\\pi} = \\{\\pi_1, \\cdots, \\pi_K\\}.", "source": "marker_v2", "marker_block_id": "/page/4/Text/10"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0041", "section": "Algorithm 1 TaskSampler", "page_start": 5, "page_end": 5, "type": "ListGroup", "text": "3: Sample the size of inputs N \\sim p_N . 4: Sample the data points \\mathbf{X} (X_1, \\dots, X_N) \\stackrel{\\text{i.i.d.}}{\\sim} p(\\cdot | \\boldsymbol{\\theta}). 5: return An (isotropic) GMM task \\mathcal{T} = (\\mathbf{X}, \\boldsymbol{\\theta}, K) .", "source": "marker_v2", "marker_block_id": "/page/4/ListGroup/249"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0042", "section": "Algorithm 1 TaskSampler", "page_start": 5, "page_end": 5, "type": "Text", "text": "Algorithm 2 (Meta) Training procedure for TGMM", "source": "marker_v2", "marker_block_id": "/page/4/Text/14"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0043", "section": "Algorithm 1 TaskSampler", "page_start": 5, "page_end": 5, "type": "Text", "text": "Require: task dimension d, task types \\mathcal{K} = \\{K_1, \\dots, K_s\\} , number of tasks n per step, number of steps T.", "source": "marker_v2", "marker_block_id": "/page/4/Text/15"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0044", "section": "Algorithm 1 TaskSampler", "page_start": 5, "page_end": 5, "type": "ListGroup", "text": "1: Initialize a TGMM model TGMM_{\\mathbf{\\Theta}^{(0)}} . 2: for t = 1 : T do 3: Sample n tasks \\{\\mathcal{T}_i\\}_{i\\in[n]} independently using the TaskSampler from Algorithm 1. 4: Compute the training objective \\widehat{L}_n\\left(\\Theta^{(t-1)}\\right) as in (2). 5: Update \\Theta^{(t-1)} into \\Theta^{(t)} using any gradient based training algorithm like AdamW. 6: end for 7: return Trained model TGMM_{\\Theta^{(T)}} .", "source": "marker_v2", "marker_block_id": "/page/4/ListGroup/250"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0045", "section": "2.3 META TRAINING PROCEDURE", "page_start": 5, "page_end": 5, "type": "Text", "text": "We adopt the meta-training framework as in Garg et al. (2022); Bai et al. (2023) and utilize diverse synthetic tasks to learn the TGMM model. In particular, during each step of the learning process, we first use a TaskSampler routine (described in Algorithm 1) to generate a batch of n tasks, with each task having a probably distinct sample size. The TGMM model outputs estimates for each task, i.e., \\{\\widehat{\\mu}_k, \\widehat{\\pi}_k\\}_{k \\in [K]} = \\mathrm{TGMM}_{\\Theta}(\\mathbf{X}; K) . Define \\widehat{\\pi} := \\{\\widehat{\\pi}_k\\}_{k \\in [K]} and \\widehat{\\boldsymbol{\\mu}} := \\{\\widehat{\\mu}_k\\}_{k \\in [K]} . For a batch of tasks \\{\\mathcal{T}_i\\}_{i \\in [n]} = \\{\\mathbf{X}_i, \\boldsymbol{\\theta}_i, K_i\\}_{i \\in [n]} , denote by \\boldsymbol{\\theta}_i = \\boldsymbol{\\mu}_i \\cup \\boldsymbol{\\pi}_i and \\widehat{\\boldsymbol{\\theta}}_i = \\widehat{\\boldsymbol{\\mu}}_i \\cup \\widehat{\\boldsymbol{\\pi}}_i = \\mathrm{TGMM}_{\\Theta}(\\mathbf{X}_i; K_i) , i \\in [n] . Then the learning objective is thus:", "source": "marker_v2", "marker_block_id": "/page/4/Text/24"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0046", "section": "2.3 META TRAINING PROCEDURE", "page_start": 5, "page_end": 5, "type": "Equation", "text": "\\widehat{L}_n(\\mathbf{\\Theta}) = \\frac{1}{n} \\sum_{i=1}^n \\ell_{\\mu}(\\widehat{\\boldsymbol{\\mu}}_i, \\boldsymbol{\\mu}_i) + \\ell_{\\pi}(\\widehat{\\boldsymbol{\\pi}}_i, \\boldsymbol{\\pi}_i). \\tag{2}", "source": "marker_v2", "marker_block_id": "/page/4/Equation/25"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0047", "section": "2.3 META TRAINING PROCEDURE", "page_start": 5, "page_end": 5, "type": "Text", "text": "where \\ell_{\\mu} and \\ell_{\\pi} are loss functions for estimation of \\mu and \\pi , respectively. We will by default use square loss for \\ell_{\\mu} and cross entropy loss for \\ell_{\\pi} . Note that the task sampling procedure relies on several sampling distributions p_{\\mu}, p_{\\pi}, p_{N}, p_{K} , which are themselves dependent upon some global configurations such as the dimension d as well as the task types \\mathcal{K} . We will omit those dependencies on global configurations when they are clear from context. The (meta) training procedure is detailed in Algorithm 2.", "source": "marker_v2", "marker_block_id": "/page/4/Text/26"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0048", "section": "3 EXPERIMENTS", "page_start": 5, "page_end": 5, "type": "Text", "text": "In this section, we empirically investigate TGMM's capability of learning to solve GMMs. We focus on the following research questions (RQ):", "source": "marker_v2", "marker_block_id": "/page/4/Text/28"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0049", "section": "3 EXPERIMENTS", "page_start": 5, "page_end": 5, "type": "Text", "text": "RQ1 Effectiveness: How well do TGMM solve GMM problems, compared to classical algorithms?", "source": "marker_v2", "marker_block_id": "/page/4/Text/29"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0050", "section": "3 EXPERIMENTS", "page_start": 5, "page_end": 5, "type": "Text", "text": "RQ2 Robustness : How well does TGMM perform over test tasks unseen during training?", "source": "marker_v2", "marker_block_id": "/page/4/Text/30"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0051", "section": "3 EXPERIMENTS", "page_start": 5, "page_end": 5, "type": "Text", "text": "RQ3 Flexibility : Can we extend the current formulation by adopting alternative backbone architectures or relaxing the isotropic setting to more sophisticated models like anisotropic GMM?", "source": "marker_v2", "marker_block_id": "/page/4/Text/31"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0052", "section": "3.1 EXPERIMENTAL SETUP", "page_start": 6, "page_end": 6, "type": "Text", "text": "Metrics. We use \\ell_2 -error as evaluation metrics in the experiments. We denote the output of the TGMM as \\widehat{\\boldsymbol{\\theta}} := \\{\\widehat{\\pi}_1, \\widehat{\\mu}_1, \\widehat{\\pi}_2, \\widehat{\\mu}_2, \\cdots \\widehat{\\pi}_K, \\widehat{\\mu}_K\\} . The rigorous definition is", "source": "marker_v2", "marker_block_id": "/page/5/Text/3"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0053", "section": "3.1 EXPERIMENTAL SETUP", "page_start": 6, "page_end": 6, "type": "Equation", "text": "\\frac{1}{K} \\sum_{k \\in [K]} \\left( \\frac{1}{d} \\left\\| \\widehat{\\mu}_{\\tilde{\\sigma}(i)} - \\mu_i \\right\\|^2 + \\left( \\widehat{\\pi}_{\\tilde{\\sigma}(i)} - \\pi_i \\right)^2 \\right),", "source": "marker_v2", "marker_block_id": "/page/5/Equation/4"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0054", "section": "3.1 EXPERIMENTAL SETUP", "page_start": 6, "page_end": 6, "type": "Text", "text": "where \\tilde{\\sigma} is the permutation such that \\tilde{\\sigma} = \\arg\\min_{\\sigma} \\sum_{k \\in [K]} \\|\\hat{\\mu}_{\\sigma(i)} - \\mu_i\\|^2 . We obtain the permutation via solving a linear assignment program using the Jonker-Volgenant algorithm(Crouse, 2016). We also report all the experimental results under two alternative metrics: cluster-classification accuracy and log-likelihood in Section H.2.", "source": "marker_v2", "marker_block_id": "/page/5/Text/5"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0055", "section": "RQ1: Effectiveness", "page_start": 6, "page_end": 6, "type": "Text", "text": "We compare the performance of a learned TGMM with the classical EM algorithm and spectral algorithm under 4 scenarios where the problem dimension ranges over \\{2,8,32,128\\} . The results are reported in Figure 2. We observe that all three algorithms perform competitively (reaching almost zero estimation error) when K=2. However, as the estimation problem gets more challenging as K increases, the EM algorithm gets trapped in local minima and underperforms both spectral and TGMM. Moreover, while the spectral algorithm performs comparably with TGMM, it cannot handle cases when K>d, which is effectively mitigated by TGMM, with corresponding performances surpassing those of the EM algorithm. This demonstrates the effectiveness of TGMM for learning an estimation algorithm that efficiently solves GMM problems.", "source": "marker_v2", "marker_block_id": "/page/5/Text/8"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0056", "section": "RQ1: Effectiveness", "page_start": 6, "page_end": 6, "type": "FigureGroup", "text": "Figure 2: Performance comparison between TGMM and two classical algorithms, reported in \\ell_2 -error.", "source": "marker_v2", "marker_block_id": "/page/5/FigureGroup/142"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0057", "section": "RQ1: Effectiveness", "page_start": 6, "page_end": 6, "type": "Text", "text": "RQ2: Robustness To assess the robustness of the learned TGMM, we consider two types of test-time distribution shifts:", "source": "marker_v2", "marker_block_id": "/page/5/Text/11"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0058", "section": "RQ1: Effectiveness", "page_start": 6, "page_end": 6, "type": "ListGroup", "text": "1. Shifts in sample size N Under this scenario, we evaluate the learned TGMM model on tasks with sample size N^{\\text{test}} that are unseen during training. 2. Shifts in sampling distributions Under this scenario, we test the learned TGMM model on tasks that are sampled from different sampling distributions that are used during training. Specifically, we use the same training sampling configuration as stated in Section 3.1 and test on the following", "source": "marker_v2", "marker_block_id": "/page/5/ListGroup/143"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0059", "section": "RQ1: Effectiveness", "page_start": 7, "page_end": 7, "type": "Text", "text": "perturbed sampling scheme, with \\tilde{\\mu}_k = \\mu_k + \\sigma_p \\varepsilon_k , where \\mu_k \\stackrel{i.i.d.}{\\sim} \\text{Unif}\\left([-5,5]^d\\right), \\varepsilon_k \\stackrel{i.i.d.}{\\sim} \\mathcal{N}(0,I_d) , k \\in [K] and \\{\\varepsilon_k\\}_{k \\in [K]} is independent with \\{\\mu_k\\}_{k \\in [K]} .", "source": "marker_v2", "marker_block_id": "/page/6/Text/1"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0060", "section": "RQ1: Effectiveness", "page_start": 7, "page_end": 7, "type": "FigureGroup", "text": "Figure 3: Assessments of TGMM under test-time task distribution shifts I: A line with N_0^{\\rm train} \\to N^{\\rm test} draws the performance of a TGMM model trained over tasks with sample size randomly sampled in [N_0^{\\rm train}/2, N_0^{\\rm train}] and evaluated over tasks with sample size N^{\\rm test} . We can view the configuration 128 \\to 128 as an in-distribution test and the rest as out-of-distribution tests.", "source": "marker_v2", "marker_block_id": "/page/6/FigureGroup/112"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0061", "section": "RQ1: Effectiveness", "page_start": 7, "page_end": 7, "type": "Text", "text": "In Figure 3, we report the assessments regarding shifts in sample size, where we set N_{\\text{test}} to be 128 and vary the training configuration N_0 to range over \\{32,64,128\\} , respectively. The results demonstrate graceful performance degradation of out-of-domain testing performance in comparison to the in-domain performance. To measure performance over shifted test-time sampling distributions, we vary the perturbation scale \\sigma_p \\in \\{0,1,\\ldots,10\\} with problem dimension fixed at d=8. The results are illustrated in Figure 4 along with comparisons to EM and spectral baselines. As shown in the results, with the increase of the perturbation scale, the estimation problem gets much harder. Nevertheless, the learned TGMM can still outperform the EM algorithm when K>2. Both pieces of evidence suggest that our meta-training procedure indeed learns an algorithm instead of overfitting to some training distribution.", "source": "marker_v2", "marker_block_id": "/page/6/Text/4"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0062", "section": "RQ1: Effectiveness", "page_start": 7, "page_end": 7, "type": "FigureGroup", "text": "Figure 4: Assessments of TGMM under test-time task distribution shifts II: \\ell_2 -error of estimation when the test-time tasks \\mathcal{T}^{\\text{test}} are sampled using a mean vector sampling distribution p_{\\mu}^{\\text{test}} different from the one used during training.", "source": "marker_v2", "marker_block_id": "/page/6/FigureGroup/113"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0063", "section": "RQ1: Effectiveness", "page_start": 7, "page_end": 7, "type": "Text", "text": "RQ3: Flexibility Finally, we initiate two studies that extend both the TGMM framework and the (meta) learning problem of solving isotropic GMMs. In our first study, we investigated alternative architectures for the TGMM backbone. Motivated by previous studies(Park et al., 2024) that demonstrate the in-context learning capability of linear attention models such as Mamba series(Gu & Dao, 2023; Dao & Gu, 2024). We test replacing the backbone of TGMM with a Mamba2(Dao & Gu, 2024) model with its detailed specifications and experimental setups listed in Section H.1. The results are reported in Figure 5, suggesting that while utilizing Mamba2 as the TGMM backbone still yields non-trivial estimation efficacy, it is in general inferior to transformer backbone under comparable model complexity.", "source": "marker_v2", "marker_block_id": "/page/6/Text/7"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0064", "section": "RQ1: Effectiveness", "page_start": 7, "page_end": 7, "type": "Text", "text": "In our second study, we adapted TGMM to be compatible with more sophisticated GMM tasks via relaxing the isotropic assumption. Specifically, we construct anisotropic GMM tasks via equipping it with another scale sampling mechanism p_{\\sigma} , where for each task we sample \\sigma \\sim \\text{softplus}(\\tilde{\\sigma}) with \\tilde{\\sigma} being sampled uniformly from [-1,1]^d . We adjust the output structure of TGMM accordingly so that its outputs can be decoded into both estimates of both mean vectors, mixture probabilities, and scales, which are detailed in Section H.1. Note that the spectral algorithm does not directly apply to anisotropic setups, limiting its flexibility. Consequently, we compare TGMM with the EM approach and plot results in Figure 6 with the \\ell_2 -error metric accommodating errors from scale estimation. The", "source": "marker_v2", "marker_block_id": "/page/6/Text/8"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0065", "section": "RQ1: Effectiveness", "page_start": 8, "page_end": 8, "type": "Text", "text": "results demonstrate a similar trend as in evaluations in the isotropic case, showcasing TGMM as a versatile tool in GMM learning problems.", "source": "marker_v2", "marker_block_id": "/page/7/Text/1"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0066", "section": "RQ1: Effectiveness", "page_start": 8, "page_end": 8, "type": "Text", "text": "Additional experiments We postpone some further evaluations to Section H, where we present a complete report consisting of more metrics and conduct several ablations on the effects of backbone scales and sample sizes.", "source": "marker_v2", "marker_block_id": "/page/7/Text/2"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0067", "section": "RQ1: Effectiveness", "page_start": 8, "page_end": 8, "type": "FigureGroup", "text": "Figure 5: Performance comparisons between TGMM using transformer and Mamba2 as backbone, reported in \\ell_2 -error.", "source": "marker_v2", "marker_block_id": "/page/7/FigureGroup/114"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0068", "section": "RQ1: Effectiveness", "page_start": 8, "page_end": 8, "type": "FigureGroup", "text": "Figure 6: Performance comparison between TGMM and the EM algorithm on anisotropic GMM tasks, reported in \\ell_2 -error.", "source": "marker_v2", "marker_block_id": "/page/7/FigureGroup/115"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0069", "section": "RQ1: Effectiveness", "page_start": 8, "page_end": 8, "type": "Text", "text": "Remark 1. One might be concerned with the fairness of comparisons between TGMM pre-training and EM/spectral method. We would like to point out that the only additional information that TGMM receives during meta-training is the (implicitly provided) distributional information. The empirical results show that TGMM can generalize beyond the meta-training distribution.", "source": "marker_v2", "marker_block_id": "/page/7/Text/7"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0070", "section": "4 THEORETICAL UNDERSTANDINGS", "page_start": 8, "page_end": 8, "type": "Text", "text": "In this section, we provide some theoretical understandings for the experiments.", "source": "marker_v2", "marker_block_id": "/page/7/Text/9"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0071", "section": "4.1 Understanding TGMM", "page_start": 8, "page_end": 8, "type": "Text", "text": "We investigate the expressive power of transformers-for-Gaussian-mixtures(TGMM) as demonstrated in Section 3. Our analysis presents two key findings that elucidate the transformer's effectiveness for GMM estimation: 1. Transformer can approximate the EM algorithm; 2. Transformer can approximate the power iteration of cubic tensor.", "source": "marker_v2", "marker_block_id": "/page/7/Text/11"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0072", "section": "4.1 Understanding TGMM", "page_start": 8, "page_end": 8, "type": "Text", "text": "Transformer can approximate the EM algorithm. We show that transformer can efficiently approximate the EM algorithm (Algorithm B.1; see Section B) and estimate the parameters of GMM. Moreover, we show that transformer with one backbone can handle tasks with different dimensions and components simultaneously. The formal statement appears in Section F due to space limitations.", "source": "marker_v2", "marker_block_id": "/page/7/Text/12"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0073", "section": "4.1 Understanding TGMM", "page_start": 8, "page_end": 8, "type": "Text", "text": "Theorem 1 (Informal). There exists a 2L-layer transformer \\mathrm{TF}_{\\Theta} such that for any d \\leq d_0 , K \\leq K_0 and task \\mathcal{T} = (\\mathbf{X}, \\boldsymbol{\\theta}, K) satisfying some regular conditions, given suitable embeddings, \\mathrm{TF}_{\\Theta} approximates EM algorithm L steps and estimates \\boldsymbol{\\theta} efficiently.", "source": "marker_v2", "marker_block_id": "/page/7/Text/13"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0074", "section": "4.1 Understanding TGMM", "page_start": 8, "page_end": 8, "type": "Text", "text": "Transformer can approximate power iteration of cubic tensor. Since directly implementing the spectral algorithm with transformers proves prohibitively complex, we instead demonstrate that transformers can effectively approximate its core computational step—the power iteration for cubic tensors (Algorithm 1 in Anandkumar et al. (2014); see Section B). Specifically, we prove that a single-layer transformer can approximate the iteration step:", "source": "marker_v2", "marker_block_id": "/page/7/Text/14"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0075", "section": "4.1 Understanding TGMM", "page_start": 8, "page_end": 8, "type": "Equation", "text": "v^{(j+1)} = T(I, v^{(j)}, v^{(j)}), j \\in \\mathbb{N}, (3)", "source": "marker_v2", "marker_block_id": "/page/7/Equation/15"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0076", "section": "4.1 Understanding TGMM", "page_start": 8, "page_end": 8, "type": "Text", "text": "where I denotes the identity matrix and T represents the given cubic tensor. For technical tractability, we assume the attention layer employs a ReLU activation function. The formal statement appears in Section G due to space limitations.", "source": "marker_v2", "marker_block_id": "/page/7/Text/16"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0077", "section": "4.1 Understanding TGMM", "page_start": 9, "page_end": 9, "type": "Text", "text": "Theorem 2 (Informal). There exists a 2L-layer transformer TF_{\\Theta} with ReLU activation such that for any d \\leq d_0 , T \\in \\mathbb{R}^{d \\times d \\times d} and v^{(0)} \\in \\mathbb{R}^d , given suitable embeddings, TF_{\\Theta} implements L steps of (3) exactly.", "source": "marker_v2", "marker_block_id": "/page/8/Text/1"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0078", "section": "4.1 Understanding TGMM", "page_start": 9, "page_end": 9, "type": "Text", "text": "We give some discussion of the theorems in the following remarks.", "source": "marker_v2", "marker_block_id": "/page/8/Text/2"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0079", "section": "4.1 Understanding TGMM", "page_start": 9, "page_end": 9, "type": "Text", "text": "Remark 2. (1) Theorem 1 demonstrates that a transformer architecture can approximate the EM algorithm for GMM tasks with varying numbers of components using a single shared set of parameters (i.e., one backbone \\Theta ). This finding supports the empirical effectiveness of TGMM ( RQ1 in Section 3.2). Additionally, Theorem 2 establishes that transformers can approximate power iterations for third-order tensors across different dimensions, further corroborating the model's ability to generalize across GMMs with varying component counts.", "source": "marker_v2", "marker_block_id": "/page/8/Text/3"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0080", "section": "4.1 Understanding TGMM", "page_start": 9, "page_end": 9, "type": "Text", "text": "(2) Theorem 1 holds uniformly over sample sizes N and sampling distributions under mild regularity conditions, aligning with the observed robustness of TGMM ( RQ2 in Section 3.2).", "source": "marker_v2", "marker_block_id": "/page/8/Text/4"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0081", "section": "4.1 Understanding TGMM", "page_start": 9, "page_end": 9, "type": "Text", "text": "Remark 3. Different \"readout\" functions are also required to extract task-specific parameters in our theoretical analysis, aligning with the architectural design described in Section 2.2. For further discussion, refer to Remark F.3 in Section F.2.", "source": "marker_v2", "marker_block_id": "/page/8/Text/5"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0082", "section": "4.2 PROOF IDEAS", "page_start": 9, "page_end": 9, "type": "Text", "text": "Proof Idea of Theorem 1. We present a brief overview of the proof strategy for Theorem 1. Our approach combines three key components: (1) the convergence properties of the population-EM algorithm(Kwon & Caramanis, 2020), (2) concentration bounds between population and sample quantities (established via classical empirical process theory), and (3) a novel transformer architecture construction. The transformer design is specifically motivated by the weighting properties of the softmax activation function, which naturally aligns with the EM algorithm's update structure. For intuitive understanding, Figure 7 provides a graphical illustration of this construction. The full proof is in Section F.", "source": "marker_v2", "marker_block_id": "/page/8/Text/7"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0083", "section": "4.2 PROOF IDEAS", "page_start": 9, "page_end": 9, "type": "FigureGroup", "text": "Figure 7: (Informal version)Transformer Construction for Approximating EM Algorithm Iterations. The word \"clean\" means setting all positions of the corresponding vector to zero.", "source": "marker_v2", "marker_block_id": "/page/8/FigureGroup/148"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0084", "section": "4.2 PROOF IDEAS", "page_start": 9, "page_end": 9, "type": "Text", "text": "Proof Idea of Theorem 2. To approximate (3), we perform a two-dimensional computation within a single-layer transformer. The key idea is to leverage the number of attention heads M to handle one dimension while utilizing the Q, K, V structure in the attention layer. Specifically, let T = (T_{i,j,m})_{i,j,m\\in[d]} and v^{(j)} = (v_i^{(j)})_{i\\in[d]} . Then, (3) can be rewritten as v^{(j+1)} = \\sum_{j,m\\in[d]} v_j v_l T_{:,j,m} , where T_{:,j,m} = (T_{i,j,m})_{i\\in[d]} \\in \\mathbb{R}^d . This operation can be implemented using d attention heads, where each head processes a dimension of size d (Figure 8). The complete construction and proof are provided in Section G.", "source": "marker_v2", "marker_block_id": "/page/8/Text/10"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0085", "section": "4.2 PROOF IDEAS", "page_start": 9, "page_end": 9, "type": "Equation", "text": "\\tilde{\\mathbf{h}}_i = \\mathbf{h}_i + \\frac{1}{d} \\sum_{m=1}^d \\sum_{j=1}^d \\sigma \\Big( \\left\\langle \\begin{array}{c} \\mathbf{Q}_m \\mathbf{h}_i \\\\ \\mathbf{V}_m \\mathbf{h}_j \\end{array} \\right\\rangle \\Big) \\begin{array}{c} \\mathbf{V}_m \\mathbf{h}_j \\end{array} \\longrightarrow v^{j+1} = \\sum_{m=1}^d \\begin{array}{c} \\mathbf{v}_m \\\\ \\sum_{j=1}^d \\end{array} \\Big( \\begin{array}{c} \\mathbf{v}_m \\\\ \\mathbf{v}_j \\end{array} \\Big) \\begin{array}{c} T_{:,j,m} \\\\ T_{:,j,m} \\end{array}", "source": "marker_v2", "marker_block_id": "/page/8/Equation/11"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0086", "section": "4.2 PROOF IDEAS", "page_start": 9, "page_end": 9, "type": "Caption", "text": "Figure 8: Illustration of implementing (3) via a multi-head attention structure, where colored boxes denote corresponding implementation components. Here \\sigma denotes the ReLU function.", "source": "marker_v2", "marker_block_id": "/page/8/Caption/12"}
+{"paper_id": "4hKNGmjXVQ", "chunk_id": "4hKNGmjXVQ:0087", "section": "5 CONCLUSION AND DISCUSSIONS", "page_start": 9, "page_end": 9, "type": "Text", "text": "In this paper, we investigate the capabilities of transformers in GMM tasks from both theoretical and empirical perspectives. Our work is among the earliest studies to investigate the mechanism of transformers in unsupervised learning settings. Our results establish fundamental theoretical guarantees that Transformers can efficiently implement classical algorithms—such as the EM algorithm and spectral methods. This is consistent with our empirical finding that the performance of our meta-training algorithm can interpolate between EM and the spectral method. It also opens a room for future improvement of attention-based meta-training algorithms in a broader class of unsupervised learning problems. We discuss the limitations and potential future research directions in Section E.", "source": "marker_v2", "marker_block_id": "/page/8/Text/14"}

iclr26/4hKNGmjXVQ/marker_meta.json

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+{
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+      "title": "TRANSFORMERS AS UNSUPERVISED LEARNING ALGO-\nRITHMS:",
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+      "title": "1 INTRODUCTION",
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+      "title": "2 METHODOLOGY",
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+    {
+      "title": "A LITERATURE ON DENSITY ESTIMATION USING LLMS",
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+      "title": "\u0412\nALGORITHM DETAILS",
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+      "title": "FULL NOTATION OF NETWORK ARCHITECTURE",
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+[p. 1 | section: ABSTRACT | type: Text]
+The transformer architecture has demonstrated remarkable capabilities in modern artificial intelligence, among which the capability of implicitly learning an internal model during inference time is widely believed to play a key role in the understanding of pre-trained large language models. However, most recent works have been focusing on studying supervised learning topics such as in-context learning, leaving the field of unsupervised learning largely unexplored. This paper investigates the capabilities of transformers in solving Gaussian Mixture Models (GMMs), a fundamental unsupervised learning problem through the lens of statistical estimation. We propose a transformer-based learning framework called Transformer for Gaussian Mixture Models (TGMM) that simultaneously learns to solve multiple GMM tasks using a shared transformer backbone. The learned models are empirically demonstrated to effectively mitigate the limitations of classical methods such as Expectation-Maximization (EM) or spectral algorithms, at the same time exhibit reasonable robustness to distribution shifts. Theoretically, we prove that transformers can efficiently approximate both the Expectation-Maximization (EM) algorithm and a core component of spectral methods—namely, cubic tensor power iterations. These results not only improve upon prior work on approximating the EM algorithm, but also provide, to our knowledge, the first theoretical guarantee that transformers can approximate high-order tensor operations. Our study bridges the gap between practical success and theoretical understanding, positioning transformers as versatile tools for unsupervised learning. 1
+
+[p. 1 | section: 1 INTRODUCTION | type: Text]
+Large Language Models (LLMs) have achieved remarkable success across various tasks in recent years. Transformers (Vaswani et al., 2017) , the dominant architecture in modern LLMs (Brown et al., 2020) , outperform many other neural network models in efficiency and scalability. Beyond language tasks, transformers have also demonstrated strong performance in other domains, such as computer vision (Han et al., 2023; Khan et al., 2022) and reinforcement learning (Li et al., 2023a) . Given their practical success, understanding the mechanisms behind transformers has attracted growing research interest. Existing studies often treat transformers as algorithmic toolboxes, investigating their ability to implement diverse algorithms (Von Oswald et al., 2023; Bai et al., 2023; Lin et al., 2024; Giannou et al., 2025; Teh et al., 2025) –a perspective linked to meta-learning (Hospedales et al., 2021) .
+
+[p. 1 | section: 1 INTRODUCTION | type: Text]
+However, most research has focused on supervised learning settings, such as regression (Bai et al., 2023) and classification (Giannou et al., 2025) , leaving the unsupervised learning paradigm relatively unexplored. Since transformer models are typically trained in a supervised manner, unsupervised learning poses inherent challenges for transformers due to the absence of labeled data. Moreover, given the abundance of unlabeled data in real-world scenarios, investigating the mechanisms of
+
+[p. 1 | section: 1 INTRODUCTION | type: Footnote]
+1 Shanghai Center for Mathematical Sciences, Fudan University,
+
+[p. 1 | section: 1 INTRODUCTION | type: Footnote]
+1 Code available at
+
+[p. 2 | section: 1 INTRODUCTION | type: Text]
+transformers in unsupervised learning holds significant implications for practical applications. The Gaussian mixture model (GMM) represents one of the most fundamental unsupervised learning tasks in statistics, with a rich historical background (DAY, 1969; Aitkin & Wilson, 1980) and ongoing research interest (Zhang et al., 2021; Manduchi et al., 2021; Löffler et al., 2021; Ndaoud, 2022; Gribonval et al., 2021; Yu et al., 2021) . Two primary algorithmic approaches are existing for solving GMM problems: (1) likelihood-based methods employing the Expectation-Maximization (EM) algorithm (Dempster et al., 1977; Balakrishnan et al., 2017) , and (2) moment-based methods utilizing spectral algorithms (Hsu & Kakade, 2013; Anandkumar et al., 2014) . However, both algorithms have inherent limitations. The EM algorithm is prone to convergence at local optima and is highly sensitive to initialization (Moitra, 2018; Jin et al., 2016) . In contrast, while the spectral method is independent of initialization, it requires the number of components to be smaller than the data's dimensionality—an assumption that restricts its applicability to problems involving many components in low-dimensional GMMs (Hsu & Kakade, 2013) .
+
+[p. 2 | section: 1 INTRODUCTION | type: Text]
+In this work, we explore transformers for GMM parameter estimation to address two questions. (i) Can Transformers provably work for GMM in-context? (ii) Can Transformers empirically overcome the drawbacks of both EM algorithm and the spectral method? Our answers are affirmative. We find that meta-trained transformers exhibit strong performance on GMM tasks without the aforementioned limitations. Notably, we construct transformer-based solvers that efficiently solve GMMs with varying component counts simultaneously. The experimental phenomena are further backed up by novel theoretical establishments: We prove that transformers can effectively learn GMMs with different components by approximating both the EM algorithm and a key component of spectral methods on GMM tasks.
+
+[p. 2 | section: Main Contributions. | type: ListGroup]
+We propose the TGMM framework that utilizes transformers to solve multiple GMM tasks with varying numbers of components simultaneously during inference time. Through extensive experimentation, the learned TGMM model is demonstrated to achieve competitive and robust performance over synthetic GMM tasks. Notably, TGMM outperforms the popular EM algorithm in terms of estimation quality, and approximately matches the strong performance of spectral methods while enjoying better flexibility. We establish theoretical foundations by proving that transformers can approximate both the EM algorithm and a key component of spectral methods. Our approximation of the EM algorithm fundamentally leverages the weighted averaging property inherent in softmax attention, enabling simultaneous approximation of both the E and M steps. Notably, our approximation results also hold across varying dimensions and mixture components in GMM. We proved that transformers (with RELU activation) can implement cubic tensor power iterationsa crucial component of spectral algorithms for GMM. The proof is highly dependent on the multi-head structure of transformers. To the best of our knowledge, this is the first theoretical demonstration of transformers' capacity for high-order tensor calculations.
+
+[p. 2 | section: Main Contributions. | type: Text]
+Related works. Recent research has explored the mechanisms by which transformers can implement various supervised learning algorithms. For instance, Akyürek et al. (2023) , Von Oswald et al. (2023) , and Bai et al. (2023) demonstrate that transformers can perform gradient descent for linear regression problems in-context. Lin et al. (2024) shows that transformers are capable of implementing Upper Confidence Bound (UCB) algorithms, as well as other classical algorithms in reinforcement learning tasks. Giannou et al. (2025) reveals that transformers can execute in-context Newton's method for logistic regression problems. Teh et al. (2025) illustrates that transformers can approximate Robbins' estimator and solve Naive Bayes problems. Kim et al. (2024) studies the minimax optimality of transformers on nonparametric regression. Some literature on density estimation using LLMs is discussed in Section A.
+
+[p. 2 | section: Main Contributions. | type: Text]
+Comparison with prior theoretical works in unsupervised learning setting. Several recent studies have investigated the mechanisms of transformer-based models in mixture model settings (He et al., 2025a; Jin et al., 2024; He et al., 2025b) . Among these, He et al. (2025a) establishes that transformers can implement Principal Component Analysis (PCA) and leverages this to GMM clustering. However, their analysis is limited to the two-component case, restricting its broader applicability.
+
+[p. 2 | section: Main Contributions. | type: Text]
+The paper Jin et al. (2024) investigates the in-context learning capabilities of transformers for mixture linear models, a setting that differs from ours. Furthermore, their approximation construction of the transformer is limited to two-component GMMs, leaving the general case unaddressed. While
+
+[p. 3 | section: Main Contributions. | type: Text]
+they assume ReLU as the activation function—contrary to the conventional choice of softmax—their theoretical proofs rely on a key lemma from prior work Pathak et al. (2024) that assumes softmax activation, thereby introducing an inconsistency in their assumptions. The paper He et al. (2025b) studies the performance of transformers on multi-class GMM clustering, a setting closely related to ours. However, our work focuses on parameter estimation rather than clustering . We give a discussion of our theoretical improvements over their work in detail in the following paragraph. From an empirical perspective, their experiments are conducted on a small-scale transformer, which fails to validate their theoretical claims.
+
+[p. 3 | section: Main Contributions. | type: Text]
+Sharpness of our results. Our theoretical analysis fully leverages key architectural components of Transformers: the query-key-value mechanism, multi-head attention, and the properties of the activation function. It is worth pointing out that our result improves the prior work for EM approximation in several points: First, Our analysis shows that Transformers can approximate L-step EM algorithms with just O(L) layers, a significant improvement over prior work (He et al., 2025b), which requires O(KL) layers (dependent on the number of components K). Second, unlike He et al. (2025b), which needs number of attention heads M \to +\infty to get valid bounds, our results hold with M = O(1), aligning better with real-world designs. Third, our approximation bounds scale polynomially in dimension d, unlike He et al. (2025b)'s exponential dependence—a crucial improvement for high-dimensional settings. We believe our results and proofs can offer profound insights for subsequent theoretical research on transformers.
+
+[p. 3 | section: Main Contributions. | type: Text]
+Organization . The rest of paper is organized as follows. In Section 2, some background knowledge is introduced. In Section 3, we present the experimental details and findings. The theoretical results are proposed in Section 4, and some discussions are given in Section 5. The proofs and additional experimental results are given in the appendix.
+
+[p. 3 | section: Main Contributions. | type: Text]
+Notations. We introduce the following notations. Let [n] := \{1, 2, \dots, n\} . All vectors are represented as column vectors unless otherwise specified. For a vector v \in \mathbb{R}^d , we denote ||v|| as its Euclidean norm. For two sequences a_n and b_n indexed by n, we denote a_n = O(b_n) if there exists a universal constant C such that a_n \leq Cb_n for sufficiently large n.
+
+[p. 3 | section: 2.1 PRELIMINARIES | type: Text]
+The Gaussian mixture model (GMM) is a cornerstone of unsupervised learning in statistics, with deep historical roots and enduring relevance in modern research. Since its early formalizations (DAY, 1969; Aitkin & Wilson, 1980), GMM has remained a fundamental tool for clustering and density estimation, widely applied across diverse domains. Recent advances have further explored the theoretical foundations of Gaussian Mixture Models (GMMs)(Löffler et al., 2021; Ndaoud, 2022; Gribonval et al., 2021), extended their applications in incomplete data settings (Zhang et al., 2021), and integrated them with deep learning frameworks (Manduchi et al., 2021; Yu et al., 2021). Due to their versatility and interpretability, GMMs remain indispensable in unsupervised learning, effectively bridging classical statistical principles with modern machine learning paradigms. We consider the (unit-variance) isotropic Gaussian Mixture Model with K components, with its probability density function as
+
+[p. 3 | section: 2.1 PRELIMINARIES | type: Equation]
+p(x|\boldsymbol{\theta}) = \sum_{k=1}^{K} \pi_k \phi(x; \mu_k) , \qquad (1)
+
+[p. 3 | section: 2.1 PRELIMINARIES | type: Text]
+where \phi(x;\mu) is the standard Gaussian kernel, i.e. \phi(x;\mu) = \frac{1}{(2\pi)^{d/2}} \exp\left(-\frac{1}{2}(x-\mu)^{\top}(x-\mu)\right) . The parameter \boldsymbol{\theta} is defined as \boldsymbol{\theta} = \boldsymbol{\pi} \cup \boldsymbol{\mu} , where \boldsymbol{\pi} := \{\pi_1, \pi_2, \cdots, \pi_K\}, \, \pi_k \in \mathbb{R} and \boldsymbol{\mu} = \{\mu_1, \mu_2, \cdots, \mu_K\}, \mu_k \in \mathbb{R}^d, \, k \in [K] . We take N samples \mathbf{X} = \{X_i\}_{i \in [N]} from model (1). \{X_i\}_{i \in [N]} can be also rewritten as
+
+[p. 3 | section: 2.1 PRELIMINARIES | type: Equation]
+X_i = \mu_{y_i} + Z_i,
+
+[p. 3 | section: 2.1 PRELIMINARIES | type: Text]
+where \{y_i\}_{i\in[N]} are i.i.d. discrete random variables with \mathbb{P}(y=k)=\pi_k for k\in[K] and \{Z_i\}_{i\in[N]} are i.i.d. standard Gaussian random vector in \mathbb{R}^d .
+
+[p. 3 | section: 2.1 PRELIMINARIES | type: Text]
+The EM algorithm(Dempster et al., 1977) remains the most widely used approach for GMM parameter estimation. Due to space constraints, we propose the algorithm in Section B. Alternatively, the
+
+[p. 4 | section: 2.1 PRELIMINARIES | type: Text]
+spectral algorithm(Hsu & Kakade, 2013) offers an efficient moment-based approach that estimates parameters through low-order observable moments. A key component of this method is cubic tensor decomposition(Anandkumar et al., 2014). For brevity, we defer the algorithmic details to Section B.
+
+[p. 4 | section: 2.1 PRELIMINARIES | type: Text]
+Next, we give a rigorous definition of the transformer model. To maintain consistency with existing literature, we adopt the notational conventions presented in Bai et al. (2023), with modifications tailored to our specific context. We consider a sequence of N input vectors \{h_i\}_{i=1}^N \subset \mathbb{R}^D , which can be compactly represented as an input matrix \mathbf{H} = [h_1, \dots, h_N] \in \mathbb{R}^{D \times N} , where each h_i corresponds to a column of \mathbf{H} (also referred to as a token).
+
+[p. 4 | section: 2.1 PRELIMINARIES | type: Text]
+Here we introduce several useful definitions and their full notations are given in Appendix C.
+
+[p. 4 | section: 2.1 PRELIMINARIES | type: Text]
+Definition 1 (Attention layer). A (self-)attention layer with M heads is denoted as \operatorname{Attn}_{\Theta_{\operatorname{attn}}}(\cdot) with parameters \Theta_{\operatorname{attn}} = \{(\mathbf{V}_m, \mathbf{Q}_m, \mathbf{K}_m)\}_{m \in [M]} \subset \mathbb{R}^{D \times D} .
+
+[p. 4 | section: 2.1 PRELIMINARIES | type: Text]
+Definition 2 (MLP layer). A (token-wise) MLP layer with hidden dimension D' is denoted as \mathrm{MLP}_{\mathbf{\Theta}_{\mathtt{nlp}}}(\cdot) with parameters \mathbf{\Theta}_{\mathtt{nlp}} = (\mathbf{W}_1, \mathbf{W}_2) \in \mathbb{R}^{D' \times D} \times \mathbb{R}^{D \times D'} .
+
+[p. 4 | section: 2.1 PRELIMINARIES | type: Text]
+Definition 3 (Transformer). An L-layer transformer, denoted as \mathrm{TF}_{\Theta_{\mathrm{TF}}}(\cdot) , is a composition of L self-attention layers each followed by an MLP layer:
+
+[p. 4 | section: 2.1 PRELIMINARIES | type: Equation]
+\mathrm{TF}_{\boldsymbol{\Theta}_{\mathrm{TF}}}(\mathbf{H}) = \mathrm{MLP}_{\boldsymbol{\Theta}_{\mathrm{nlp}}^{(L)}} \Big( \mathrm{Attn}_{\boldsymbol{\Theta}_{\mathrm{attn}}^{(L)}} \Big( \cdots \mathrm{MLP}_{\boldsymbol{\Theta}_{\mathrm{nlp}}^{(1)}} \Big( \mathrm{Attn}_{\boldsymbol{\Theta}_{\mathrm{attn}}^{(1)}} (\mathbf{H}) \Big) \Big) \Big).
+
+[p. 4 | section: 2.2 THE TGMM ARCHITECTURE | type: Text]
+A recent line of work(Xie et al., 2021; Garg et al., 2022; Bai et al., 2023; Akyürek et al., 2023; Li et al., 2023b) has been studying the capability of transformer that functions as a data-driven algorithm under the context of in-context learning (ICL). However, in contrast to the setups therein where inputs consist of both features and labels, under the unsupervised GMM setup, there is no explicitly provided label information. Therefore, we formulate the learning problem as learning an estimation algorithm instead of learning a prediction algorithm as in the case of ICL. A notable property of GMM is that the structure of the estimand depends on an unknown parameter K, which is often treated as a hyper-parameter in GMM estimation(Titterington et al., 1985; McLachlan & Peel, 2000). For clarity of representation, we define an isotropic Gaussian mixture task as \mathcal{T} = (\theta, \mathbf{X}, K) , where \mathbf{X} is a i.i.d. sample generated according to ground truth \theta according to the isotropic GMM law and K is the configuration used during estimation which we assume to be the same as the number of components of the ground truth \theta . The GMM task is solved via applying some algorithm \mathcal{A} that takes \mathbf{X} and K as inputs and outputs an estimate of the ground truth \hat{\theta} = \mathcal{A}(\mathbf{X}; K) .
+
+[p. 4 | section: 2.2 THE TGMM ARCHITECTURE | type: Text]
+In this paper, we propose a transformer-based architecture, transformers-for-Gaussian-mixtures (TGMM), as a GMM task solver that allows flexibility in its outputs, while at the same time being parameter-efficient, as illustrated in Figure 1: A TGMM model supports solving s different GMM tasks with K \in \mathcal{K} := \{K_1, \ldots, K_s\} . Given inputs N data points \mathbf{X} \in \mathbb{R}^{d \times N} and a structure configuration of the estimand K. TGMM first augments the inputs with auxiliary configurations about K via concatenating it with a task embedding \mathbf{P} = \text{embed}(K) , i.e., \mathbf{H} = [\mathbf{X}||\mathbf{P}] , and use a linear Readin layer to project the augmented inputs onto a shared hidden representation space for several estimand structures \{K_1, \ldots, K_s\} , which is then manipulated by a shared transformer backbone that produces task-aware hidden representations. The TGMM estimates are then decoded by task-specific Readout modules. More precisely, with target decoding parameters of K components, the Readout module first performs an attentive-pooling operation (Lee et al., 2019):
+
+[p. 4 | section: 2.2 THE TGMM ARCHITECTURE | type: Equation]
+\mathbf{O} = (\mathbf{V}_o \mathbf{H}) \mathrm{SoftMax} ((\mathbf{K}_o \mathbf{H})^{\top} \mathbf{Q}_o) \in \mathbb{R}^{(d+K) \times K},
+
+[p. 4 | section: 2.2 THE TGMM ARCHITECTURE | type: Text]
+where \mathbf{V}_o, \mathbf{K}_o \in \mathbb{R}^{(d+K)\times D} , \mathbf{Q}_o \in \mathbb{R}^{(d+K)\times K} . The estimates for mixture probability are then extracted by a row-wise mean-pooling of the first K rows of \mathbf{O} , and the estimates for mean vectors are the last d rows of \mathbf{O} . We wrap the above procedure as \{\widehat{\pi}_k, \widehat{\mu}_k\}_{i\in[K]} = \operatorname{Readout}_{\Theta_{\text{out}}}(\mathbf{H}) . TGMM is parameter-efficient in the sense that it only introduces extra parameter complexities of the order O(sdD) in addition to the backbone. We give a more detailed explanation of the parameter efficiency of TGMM in appendix Section D. We wrap the TGMM model into the following form:
+
+[p. 4 | section: 2.2 THE TGMM ARCHITECTURE | type: Equation]
+\mathrm{TGMM}_{\mathbf{\Theta}}(\mathbf{X}; K) = \mathrm{Readout}_{\mathbf{\Theta}_{out}}(\mathrm{TF}_{\mathbf{\Theta}_{TF}}(\mathrm{Readin}_{\mathbf{\Theta}_{in}}([\mathbf{X}||\mathrm{embed}(K)]))).
+
+[p. 4 | section: 2.2 THE TGMM ARCHITECTURE | type: Text]
+Above, the parameter \Theta = (\Theta_{TF}, \Theta_{in}, \Theta_{out}) consists of the parameters in the transformer \Theta_{TF} and the parameters in the Readin and the Readout functions \Theta_{in}, \Theta_{out} .
+
+[p. 5 | section: 2.2 THE TGMM ARCHITECTURE | type: FigureGroup]
+Figure 1: Illustration of the proposed TGMM architecture: TGMM utilizes a shared transformer backbone that supports solving s different kind of GMM tasks via a task-specific Readout strategies.
+
+[p. 5 | section: Algorithm 1 TaskSampler | type: Text]
+Require: sampling
+
+[p. 5 | section: Algorithm 1 TaskSampler | type: Text]
+distributions
+
+[p. 5 | section: Algorithm 1 TaskSampler | type: Text]
+p_{\mu}, p_{\pi}, p_N, p_K .
+
+[p. 5 | section: Algorithm 1 TaskSampler | type: ListGroup]
+1: Sample the type of task (i.e., number of mixture components) K \sim p_K . Sample a GMM task according to the type of task
+
+[p. 5 | section: Algorithm 1 TaskSampler | type: Equation]
+\bm{\theta} = (\bm{\mu}, \bm{\pi}), \bm{\mu} \sim p_{\mu}, \bm{\pi} \sim p_{\pi},
+
+[p. 5 | section: Algorithm 1 TaskSampler | type: Text]
+where \boldsymbol{\mu} = \{\mu_1, \cdots, \mu_K\}, \boldsymbol{\pi} = \{\pi_1, \cdots, \pi_K\}.
+
+[p. 5 | section: Algorithm 1 TaskSampler | type: ListGroup]
+3: Sample the size of inputs N \sim p_N . 4: Sample the data points \mathbf{X} (X_1, \dots, X_N) \stackrel{\text{i.i.d.}}{\sim} p(\cdot | \boldsymbol{\theta}). 5: return An (isotropic) GMM task \mathcal{T} = (\mathbf{X}, \boldsymbol{\theta}, K) .
+
+[p. 5 | section: Algorithm 1 TaskSampler | type: Text]
+Algorithm 2 (Meta) Training procedure for TGMM
+
+[p. 5 | section: Algorithm 1 TaskSampler | type: Text]
+Require: task dimension d, task types \mathcal{K} = \{K_1, \dots, K_s\} , number of tasks n per step, number of steps T.
+
+[p. 5 | section: Algorithm 1 TaskSampler | type: ListGroup]
+1: Initialize a TGMM model TGMM_{\mathbf{\Theta}^{(0)}} . 2: for t = 1 : T do 3: Sample n tasks \{\mathcal{T}_i\}_{i\in[n]} independently using the TaskSampler from Algorithm 1. 4: Compute the training objective \widehat{L}_n\left(\Theta^{(t-1)}\right) as in (2). 5: Update \Theta^{(t-1)} into \Theta^{(t)} using any gradient based training algorithm like AdamW. 6: end for 7: return Trained model TGMM_{\Theta^{(T)}} .
+
+[p. 5 | section: 2.3 META TRAINING PROCEDURE | type: Text]
+We adopt the meta-training framework as in Garg et al. (2022); Bai et al. (2023) and utilize diverse synthetic tasks to learn the TGMM model. In particular, during each step of the learning process, we first use a TaskSampler routine (described in Algorithm 1) to generate a batch of n tasks, with each task having a probably distinct sample size. The TGMM model outputs estimates for each task, i.e., \{\widehat{\mu}_k, \widehat{\pi}_k\}_{k \in [K]} = \mathrm{TGMM}_{\Theta}(\mathbf{X}; K) . Define \widehat{\pi} := \{\widehat{\pi}_k\}_{k \in [K]} and \widehat{\boldsymbol{\mu}} := \{\widehat{\mu}_k\}_{k \in [K]} . For a batch of tasks \{\mathcal{T}_i\}_{i \in [n]} = \{\mathbf{X}_i, \boldsymbol{\theta}_i, K_i\}_{i \in [n]} , denote by \boldsymbol{\theta}_i = \boldsymbol{\mu}_i \cup \boldsymbol{\pi}_i and \widehat{\boldsymbol{\theta}}_i = \widehat{\boldsymbol{\mu}}_i \cup \widehat{\boldsymbol{\pi}}_i = \mathrm{TGMM}_{\Theta}(\mathbf{X}_i; K_i) , i \in [n] . Then the learning objective is thus:
+
+[p. 5 | section: 2.3 META TRAINING PROCEDURE | type: Equation]
+\widehat{L}_n(\mathbf{\Theta}) = \frac{1}{n} \sum_{i=1}^n \ell_{\mu}(\widehat{\boldsymbol{\mu}}_i, \boldsymbol{\mu}_i) + \ell_{\pi}(\widehat{\boldsymbol{\pi}}_i, \boldsymbol{\pi}_i). \tag{2}
+
+[p. 5 | section: 2.3 META TRAINING PROCEDURE | type: Text]
+where \ell_{\mu} and \ell_{\pi} are loss functions for estimation of \mu and \pi , respectively. We will by default use square loss for \ell_{\mu} and cross entropy loss for \ell_{\pi} . Note that the task sampling procedure relies on several sampling distributions p_{\mu}, p_{\pi}, p_{N}, p_{K} , which are themselves dependent upon some global configurations such as the dimension d as well as the task types \mathcal{K} . We will omit those dependencies on global configurations when they are clear from context. The (meta) training procedure is detailed in Algorithm 2.
+
+[p. 5 | section: 3 EXPERIMENTS | type: Text]
+In this section, we empirically investigate TGMM's capability of learning to solve GMMs. We focus on the following research questions (RQ):
+
+[p. 5 | section: 3 EXPERIMENTS | type: Text]
+RQ1 Effectiveness: How well do TGMM solve GMM problems, compared to classical algorithms?
+
+[p. 5 | section: 3 EXPERIMENTS | type: Text]
+RQ2 Robustness : How well does TGMM perform over test tasks unseen during training?
+
+[p. 5 | section: 3 EXPERIMENTS | type: Text]
+RQ3 Flexibility : Can we extend the current formulation by adopting alternative backbone architectures or relaxing the isotropic setting to more sophisticated models like anisotropic GMM?
+
+[p. 6 | section: 3.1 EXPERIMENTAL SETUP | type: Text]
+Metrics. We use \ell_2 -error as evaluation metrics in the experiments. We denote the output of the TGMM as \widehat{\boldsymbol{\theta}} := \{\widehat{\pi}_1, \widehat{\mu}_1, \widehat{\pi}_2, \widehat{\mu}_2, \cdots \widehat{\pi}_K, \widehat{\mu}_K\} . The rigorous definition is
+
+[p. 6 | section: 3.1 EXPERIMENTAL SETUP | type: Equation]
+\frac{1}{K} \sum_{k \in [K]} \left( \frac{1}{d} \left\| \widehat{\mu}_{\tilde{\sigma}(i)} - \mu_i \right\|^2 + \left( \widehat{\pi}_{\tilde{\sigma}(i)} - \pi_i \right)^2 \right),
+
+[p. 6 | section: 3.1 EXPERIMENTAL SETUP | type: Text]
+where \tilde{\sigma} is the permutation such that \tilde{\sigma} = \arg\min_{\sigma} \sum_{k \in [K]} \|\hat{\mu}_{\sigma(i)} - \mu_i\|^2 . We obtain the permutation via solving a linear assignment program using the Jonker-Volgenant algorithm(Crouse, 2016). We also report all the experimental results under two alternative metrics: cluster-classification accuracy and log-likelihood in Section H.2.
+
+[p. 6 | section: RQ1: Effectiveness | type: Text]
+We compare the performance of a learned TGMM with the classical EM algorithm and spectral algorithm under 4 scenarios where the problem dimension ranges over \{2,8,32,128\} . The results are reported in Figure 2. We observe that all three algorithms perform competitively (reaching almost zero estimation error) when K=2. However, as the estimation problem gets more challenging as K increases, the EM algorithm gets trapped in local minima and underperforms both spectral and TGMM. Moreover, while the spectral algorithm performs comparably with TGMM, it cannot handle cases when K>d, which is effectively mitigated by TGMM, with corresponding performances surpassing those of the EM algorithm. This demonstrates the effectiveness of TGMM for learning an estimation algorithm that efficiently solves GMM problems.
+
+[p. 6 | section: RQ1: Effectiveness | type: FigureGroup]
+Figure 2: Performance comparison between TGMM and two classical algorithms, reported in \ell_2 -error.
+
+[p. 6 | section: RQ1: Effectiveness | type: Text]
+RQ2: Robustness To assess the robustness of the learned TGMM, we consider two types of test-time distribution shifts:
+
+[p. 6 | section: RQ1: Effectiveness | type: ListGroup]
+1. Shifts in sample size N Under this scenario, we evaluate the learned TGMM model on tasks with sample size N^{\text{test}} that are unseen during training. 2. Shifts in sampling distributions Under this scenario, we test the learned TGMM model on tasks that are sampled from different sampling distributions that are used during training. Specifically, we use the same training sampling configuration as stated in Section 3.1 and test on the following
+
+[p. 7 | section: RQ1: Effectiveness | type: Text]
+perturbed sampling scheme, with \tilde{\mu}_k = \mu_k + \sigma_p \varepsilon_k , where \mu_k \stackrel{i.i.d.}{\sim} \text{Unif}\left([-5,5]^d\right), \varepsilon_k \stackrel{i.i.d.}{\sim} \mathcal{N}(0,I_d) , k \in [K] and \{\varepsilon_k\}_{k \in [K]} is independent with \{\mu_k\}_{k \in [K]} .
+
+[p. 7 | section: RQ1: Effectiveness | type: FigureGroup]
+Figure 3: Assessments of TGMM under test-time task distribution shifts I: A line with N_0^{\rm train} \to N^{\rm test} draws the performance of a TGMM model trained over tasks with sample size randomly sampled in [N_0^{\rm train}/2, N_0^{\rm train}] and evaluated over tasks with sample size N^{\rm test} . We can view the configuration 128 \to 128 as an in-distribution test and the rest as out-of-distribution tests.
+
+[p. 7 | section: RQ1: Effectiveness | type: Text]
+In Figure 3, we report the assessments regarding shifts in sample size, where we set N_{\text{test}} to be 128 and vary the training configuration N_0 to range over \{32,64,128\} , respectively. The results demonstrate graceful performance degradation of out-of-domain testing performance in comparison to the in-domain performance. To measure performance over shifted test-time sampling distributions, we vary the perturbation scale \sigma_p \in \{0,1,\ldots,10\} with problem dimension fixed at d=8. The results are illustrated in Figure 4 along with comparisons to EM and spectral baselines. As shown in the results, with the increase of the perturbation scale, the estimation problem gets much harder. Nevertheless, the learned TGMM can still outperform the EM algorithm when K>2. Both pieces of evidence suggest that our meta-training procedure indeed learns an algorithm instead of overfitting to some training distribution.
+
+[p. 7 | section: RQ1: Effectiveness | type: FigureGroup]
+Figure 4: Assessments of TGMM under test-time task distribution shifts II: \ell_2 -error of estimation when the test-time tasks \mathcal{T}^{\text{test}} are sampled using a mean vector sampling distribution p_{\mu}^{\text{test}} different from the one used during training.
+
+[p. 7 | section: RQ1: Effectiveness | type: Text]
+RQ3: Flexibility Finally, we initiate two studies that extend both the TGMM framework and the (meta) learning problem of solving isotropic GMMs. In our first study, we investigated alternative architectures for the TGMM backbone. Motivated by previous studies(Park et al., 2024) that demonstrate the in-context learning capability of linear attention models such as Mamba series(Gu & Dao, 2023; Dao & Gu, 2024). We test replacing the backbone of TGMM with a Mamba2(Dao & Gu, 2024) model with its detailed specifications and experimental setups listed in Section H.1. The results are reported in Figure 5, suggesting that while utilizing Mamba2 as the TGMM backbone still yields non-trivial estimation efficacy, it is in general inferior to transformer backbone under comparable model complexity.
+
+[p. 7 | section: RQ1: Effectiveness | type: Text]
+In our second study, we adapted TGMM to be compatible with more sophisticated GMM tasks via relaxing the isotropic assumption. Specifically, we construct anisotropic GMM tasks via equipping it with another scale sampling mechanism p_{\sigma} , where for each task we sample \sigma \sim \text{softplus}(\tilde{\sigma}) with \tilde{\sigma} being sampled uniformly from [-1,1]^d . We adjust the output structure of TGMM accordingly so that its outputs can be decoded into both estimates of both mean vectors, mixture probabilities, and scales, which are detailed in Section H.1. Note that the spectral algorithm does not directly apply to anisotropic setups, limiting its flexibility. Consequently, we compare TGMM with the EM approach and plot results in Figure 6 with the \ell_2 -error metric accommodating errors from scale estimation. The
+
+[p. 8 | section: RQ1: Effectiveness | type: Text]
+results demonstrate a similar trend as in evaluations in the isotropic case, showcasing TGMM as a versatile tool in GMM learning problems.
+
+[p. 8 | section: RQ1: Effectiveness | type: Text]
+Additional experiments We postpone some further evaluations to Section H, where we present a complete report consisting of more metrics and conduct several ablations on the effects of backbone scales and sample sizes.
+
+[p. 8 | section: RQ1: Effectiveness | type: FigureGroup]
+Figure 5: Performance comparisons between TGMM using transformer and Mamba2 as backbone, reported in \ell_2 -error.
+
+[p. 8 | section: RQ1: Effectiveness | type: FigureGroup]
+Figure 6: Performance comparison between TGMM and the EM algorithm on anisotropic GMM tasks, reported in \ell_2 -error.
+
+[p. 8 | section: RQ1: Effectiveness | type: Text]
+Remark 1. One might be concerned with the fairness of comparisons between TGMM pre-training and EM/spectral method. We would like to point out that the only additional information that TGMM receives during meta-training is the (implicitly provided) distributional information. The empirical results show that TGMM can generalize beyond the meta-training distribution.
+
+[p. 8 | section: 4 THEORETICAL UNDERSTANDINGS | type: Text]
+In this section, we provide some theoretical understandings for the experiments.
+
+[p. 8 | section: 4.1 Understanding TGMM | type: Text]
+We investigate the expressive power of transformers-for-Gaussian-mixtures(TGMM) as demonstrated in Section 3. Our analysis presents two key findings that elucidate the transformer's effectiveness for GMM estimation: 1. Transformer can approximate the EM algorithm; 2. Transformer can approximate the power iteration of cubic tensor.
+
+[p. 8 | section: 4.1 Understanding TGMM | type: Text]
+Transformer can approximate the EM algorithm. We show that transformer can efficiently approximate the EM algorithm (Algorithm B.1; see Section B) and estimate the parameters of GMM. Moreover, we show that transformer with one backbone can handle tasks with different dimensions and components simultaneously. The formal statement appears in Section F due to space limitations.
+
+[p. 8 | section: 4.1 Understanding TGMM | type: Text]
+Theorem 1 (Informal). There exists a 2L-layer transformer \mathrm{TF}_{\Theta} such that for any d \leq d_0 , K \leq K_0 and task \mathcal{T} = (\mathbf{X}, \boldsymbol{\theta}, K) satisfying some regular conditions, given suitable embeddings, \mathrm{TF}_{\Theta} approximates EM algorithm L steps and estimates \boldsymbol{\theta} efficiently.
+
+[p. 8 | section: 4.1 Understanding TGMM | type: Text]
+Transformer can approximate power iteration of cubic tensor. Since directly implementing the spectral algorithm with transformers proves prohibitively complex, we instead demonstrate that transformers can effectively approximate its core computational step—the power iteration for cubic tensors (Algorithm 1 in Anandkumar et al. (2014); see Section B). Specifically, we prove that a single-layer transformer can approximate the iteration step:
+
+[p. 8 | section: 4.1 Understanding TGMM | type: Equation]
+v^{(j+1)} = T(I, v^{(j)}, v^{(j)}), j \in \mathbb{N}, (3)
+
+[p. 8 | section: 4.1 Understanding TGMM | type: Text]
+where I denotes the identity matrix and T represents the given cubic tensor. For technical tractability, we assume the attention layer employs a ReLU activation function. The formal statement appears in Section G due to space limitations.
+
+[p. 9 | section: 4.1 Understanding TGMM | type: Text]
+Theorem 2 (Informal). There exists a 2L-layer transformer TF_{\Theta} with ReLU activation such that for any d \leq d_0 , T \in \mathbb{R}^{d \times d \times d} and v^{(0)} \in \mathbb{R}^d , given suitable embeddings, TF_{\Theta} implements L steps of (3) exactly.
+
+[p. 9 | section: 4.1 Understanding TGMM | type: Text]
+We give some discussion of the theorems in the following remarks.
+
+[p. 9 | section: 4.1 Understanding TGMM | type: Text]
+Remark 2. (1) Theorem 1 demonstrates that a transformer architecture can approximate the EM algorithm for GMM tasks with varying numbers of components using a single shared set of parameters (i.e., one backbone \Theta ). This finding supports the empirical effectiveness of TGMM ( RQ1 in Section 3.2). Additionally, Theorem 2 establishes that transformers can approximate power iterations for third-order tensors across different dimensions, further corroborating the model's ability to generalize across GMMs with varying component counts.
+
+[p. 9 | section: 4.1 Understanding TGMM | type: Text]
+(2) Theorem 1 holds uniformly over sample sizes N and sampling distributions under mild regularity conditions, aligning with the observed robustness of TGMM ( RQ2 in Section 3.2).
+
+[p. 9 | section: 4.1 Understanding TGMM | type: Text]
+Remark 3. Different "readout" functions are also required to extract task-specific parameters in our theoretical analysis, aligning with the architectural design described in Section 2.2. For further discussion, refer to Remark F.3 in Section F.2.
+
+[p. 9 | section: 4.2 PROOF IDEAS | type: Text]
+Proof Idea of Theorem 1. We present a brief overview of the proof strategy for Theorem 1. Our approach combines three key components: (1) the convergence properties of the population-EM algorithm(Kwon & Caramanis, 2020), (2) concentration bounds between population and sample quantities (established via classical empirical process theory), and (3) a novel transformer architecture construction. The transformer design is specifically motivated by the weighting properties of the softmax activation function, which naturally aligns with the EM algorithm's update structure. For intuitive understanding, Figure 7 provides a graphical illustration of this construction. The full proof is in Section F.
+
+[p. 9 | section: 4.2 PROOF IDEAS | type: FigureGroup]
+Figure 7: (Informal version)Transformer Construction for Approximating EM Algorithm Iterations. The word "clean" means setting all positions of the corresponding vector to zero.
+
+[p. 9 | section: 4.2 PROOF IDEAS | type: Text]
+Proof Idea of Theorem 2. To approximate (3), we perform a two-dimensional computation within a single-layer transformer. The key idea is to leverage the number of attention heads M to handle one dimension while utilizing the Q, K, V structure in the attention layer. Specifically, let T = (T_{i,j,m})_{i,j,m\in[d]} and v^{(j)} = (v_i^{(j)})_{i\in[d]} . Then, (3) can be rewritten as v^{(j+1)} = \sum_{j,m\in[d]} v_j v_l T_{:,j,m} , where T_{:,j,m} = (T_{i,j,m})_{i\in[d]} \in \mathbb{R}^d . This operation can be implemented using d attention heads, where each head processes a dimension of size d (Figure 8). The complete construction and proof are provided in Section G.
+
+[p. 9 | section: 4.2 PROOF IDEAS | type: Equation]
+\tilde{\mathbf{h}}_i = \mathbf{h}_i + \frac{1}{d} \sum_{m=1}^d \sum_{j=1}^d \sigma \Big( \left\langle \begin{array}{c} \mathbf{Q}_m \mathbf{h}_i \\ \mathbf{V}_m \mathbf{h}_j \end{array} \right\rangle \Big) \begin{array}{c} \mathbf{V}_m \mathbf{h}_j \end{array} \longrightarrow v^{j+1} = \sum_{m=1}^d \begin{array}{c} \mathbf{v}_m \\ \sum_{j=1}^d \end{array} \Big( \begin{array}{c} \mathbf{v}_m \\ \mathbf{v}_j \end{array} \Big) \begin{array}{c} T_{:,j,m} \\ T_{:,j,m} \end{array}
+
+[p. 9 | section: 4.2 PROOF IDEAS | type: Caption]
+Figure 8: Illustration of implementing (3) via a multi-head attention structure, where colored boxes denote corresponding implementation components. Here \sigma denotes the ReLU function.
+
+[p. 9 | section: 5 CONCLUSION AND DISCUSSIONS | type: Text]
+In this paper, we investigate the capabilities of transformers in GMM tasks from both theoretical and empirical perspectives. Our work is among the earliest studies to investigate the mechanism of transformers in unsupervised learning settings. Our results establish fundamental theoretical guarantees that Transformers can efficiently implement classical algorithms—such as the EM algorithm and spectral methods. This is consistent with our empirical finding that the performance of our meta-training algorithm can interpolate between EM and the spectral method. It also opens a room for future improvement of attention-based meta-training algorithms in a broader class of unsupervised learning problems. We discuss the limitations and potential future research directions in Section E.

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+Anzo Teh, Mark Jabbour, and Yury Polyanskiy. Solving empirical bayes via transformers. arXiv preprint arXiv:2502.09844 , 2025. D.M. Titterington, A.F.M. Smith, and U.E. Makov. Statistical Analysis of Finite Mixture Distributions . Wiley, New York, 1985. Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Proceedings of the 31st International Conference on Neural Information Processing Systems , NIPS'17, pp. 6000–6010, Red Hook, NY, USA, 2017. Curran Associates Inc. ISBN 9781510860964. Johannes Von Oswald, Eyvind Niklasson, Ettore Randazzo, Joao Sacramento, Alexander Mordvintsev, Andrey Zhmoginov, and Max Vladymyrov. Transformers learn in-context by gradient descent. In Andreas Krause, Emma Brunskill, Kyunghyun Cho, Barbara Engelhardt, Sivan Sabato, and Jonathan Scarlett (eds.), Proceedings of the 40th International Conference on Machine Learning , volume 202 of Proceedings of Machine Learning Research , pp. 35151–35174. PMLR, 23–29 Jul 2023. URL . Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, Rémi Louf, Morgan Funtowicz, Joe Davison, Sam Shleifer, Patrick von Platen, Clara Ma, Yacine Jernite, Julien Plu, Canwen Xu, Teven Le Scao, Sylvain Gugger, Mariama Drame, Quentin Lhoest, and Alexander M. Rush. Transformers: State-of-the-art natural language processing. In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Pro cessing: System Demonstrations , pp. 38–45, Online, October 2020. Association for Computational Linguistics. URL . Sang Michael Xie, Aditi Raghunathan, Percy Liang, and Tengyu Ma. An explanation of in-context learning as implicit bayesian inference. arXiv preprint arXiv:2111.02080 , 2021. Bin Yu, Chen Chen, Ren Qi, Ruiqing Zheng, Patrick J Skillman-Lawrence, Xiaolin Wang, Anjun Ma, and Haiming Gu. scgmai: a gaussian mixture model for clustering single-cell rna-seq data based on deep autoencoder. Briefings in bioinformatics , 22(4):bbaa316, 2021. Yi Zhang, Miaomiao Li, Siwei Wang, Sisi Dai, Lei Luo, En Zhu, Huiying Xu, Xinzhong Zhu, Chaoyun Yao, and Haoran Zhou. Gaussian mixture model clustering with incomplete data. ACM Trans. Multimedia Comput. Commun. Appl. , 17(1s), March 2021. ISSN 1551-6857. doi: 10.1145/3408318. URL . Ruofei Zhao, Yuanzhi Li, and Yuekai Sun. Statistical convergence of the EM algorithm on Gaussian mixture models. Electronic Journal of Statistics , 14(1):632 – 660, 2020. doi: 10.1214/19-EJS1660.

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