Christian Kümmerle

Christian Kümmerle, Johannes Maly

We propose a new algorithm for the problem of recovering data that adheres to multiple, heterogeneous low-dimensional structures from linear observations. Focusing on data matrices that are simultaneously row-sparse and low-rank, we propose and analyze an iteratively reweighted least squares (IRLS) algorithm that is able to leverage both structures. In particular, it optimizes a combination of non-convex surrogates for row-sparsity and rank, a balancing of which is built into the algorithm. We prove locally quadratic convergence of the iterates to a simultaneously structured data matrix in a regime of minimal sample complexity (up to constants and a logarithmic factor), which is known to be impossible for a combination of convex surrogates. In experiments, we show that the IRLS method exhibits favorable empirical convergence, identifying simultaneously row-sparse and low-rank matrices from fewer measurements than state-of-the-art methods. Code is available at https://github.com/ckuemmerle/simirls.

Christian Kümmerle, Mauro Maggioni, Sui Tang

We consider the nonlinear inverse problem of learning a transition operator $\mathbf{A}$ from partial observations at different times, in particular from sparse observations of entries of its powers $\mathbf{A},\mathbf{A}^2,\cdots,\mathbf{A}^{T}$. This Spatio-Temporal Transition Operator Recovery problem is motivated by the recent interest in learning time-varying graph signals that are driven by graph operators depending on the underlying graph topology. We address the nonlinearity of the problem by embedding it into a higher-dimensional space of suitable block-Hankel matrices, where it becomes a low-rank matrix completion problem, even if $\mathbf{A}$ is of full rank. For both a uniform and an adaptive random space-time sampling model, we quantify the recoverability of the transition operator via suitable measures of incoherence of these block-Hankel embedding matrices. For graph transition operators these measures of incoherence depend on the interplay between the dynamics and the graph topology. We develop a suitable non-convex iterative reweighted least squares (IRLS) algorithm, establish its quadratic local convergence, and show that, in optimal scenarios, no more than $\mathcal{O}(rn \log(nT))$ space-time samples are sufficient to ensure accurate recovery of a rank-$r$ operator $\mathbf{A}$ of size $n \times n$. This establishes that spatial samples can be substituted by a comparable number of space-time samples. We provide an efficient implementation of the proposed IRLS algorithm with space complexity of order $O(r n T)$ and per-iteration time complexity linear in $n$. Numerical experiments for transition operators based on several graph models confirm that the theoretical findings accurately track empirical phase transitions, and illustrate the applicability and scalability of the proposed algorithm.

Nan Huang, Christian Kümmerle, Xiang Zhang

Normalization techniques are crucial for enhancing Transformer models' performance and stability in time series analysis tasks, yet traditional methods like batch and layer normalization often lead to issues such as token shift, attention shift, and sparse attention. We propose UnitNorm, a novel approach that scales input vectors by their norms and modulates attention patterns, effectively circumventing these challenges. Grounded in existing normalization frameworks, UnitNorm's effectiveness is demonstrated across diverse time series analysis tasks, including forecasting, classification, and anomaly detection, via a rigorous evaluation on 6 state-of-the-art models and 10 datasets. Notably, UnitNorm shows superior performance, especially in scenarios requiring robust attention mechanisms and contextual comprehension, evidenced by significant improvements by up to a 1.46 decrease in MSE for forecasting, and a 4.89% increase in accuracy for classification. This work not only calls for a reevaluation of normalization strategies in time series Transformers but also sets a new direction for enhancing model performance and stability. The source code is available at https://anonymous.4open.science/r/UnitNorm-5B84.

Ipsita Ghosh, Ethan Nguyen, Christian Kümmerle

Parameter-efficient training, based on low-rank optimization, has become a highly successful tool for fine-tuning large deep-learning models. However, these methods fail at low-rank pre-training tasks where maintaining the low-rank structure and the objective remains a challenging task. We propose the Quadratic Reweighted Rank Regularizer dubbed Q3R, which leads to a novel low-rank inducing training strategy inspired by the iteratively reweighted least squares (IRLS) framework. Q3R is based on a quadratic regularizer term which majorizes a smoothed log determinant serving as rank surrogate objective. Unlike other low-rank training techniques, Q3R is able to train weight matrices with prescribed, low target ranks of models that achieve comparable predictive performance as dense models, with small computational overhead, while remaining fully compatible with existing architectures. For example, we demonstrated one experiment where we are able to truncate $60\%$ and $80\%$ of the parameters of a ViT-Tiny model with $~1.3\%$ and $~4\%$ accuracy drop in CIFAR-10 performance respectively. The efficacy of Q3R is confirmed on Transformers across both image and language tasks, including for low-rank fine-tuning.

Ipsita Ghosh, Abiy Tasissa, Christian Kümmerle

The problem of finding suitable point embedding or geometric configurations given only Euclidean distance information of point pairs arises both as a core task and as a sub-problem in a variety of machine learning applications. In this paper, we aim to solve this problem given a minimal number of distance samples. To this end, we leverage continuous and non-convex rank minimization formulations of the problem and establish a local convergence guarantee for a variant of iteratively reweighted least squares (IRLS), which applies if a minimal random set of observed distances is provided. As a technical tool, we establish a restricted isometry property (RIP) restricted to a tangent space of the manifold of symmetric rank-$r$ matrices given random Euclidean distance measurements, which might be of independent interest for the analysis of other non-convex approaches. Furthermore, we assess data efficiency, scalability and generalizability of different reconstruction algorithms through numerical experiments with simulated data as well as real-world data, demonstrating the proposed algorithm's ability to identify the underlying geometry from fewer distance samples compared to the state-of-the-art.

Ali K. Rahimian, Manish K. Govind, Subhajit Maity et al.

Vision Transformers and their variants have achieved remarkable success in diverse visual perception tasks. Despite their effectiveness, they suffer from two significant limitations. First, the quadratic computational complexity of multi-head self-attention (MHSA), which restricts scalability to large token counts, and second, a high dependency on large-scale training data to attain competitive performance. In this paper, to address these challenges, we propose a novel sparse self-attention mechanism named Fibottention. Fibottention employs structured sparsity patterns derived from the Wythoff array, enabling an $\mathcal{O}(N \log N)$ computational complexity in self-attention. By design, its sparsity patterns vary across attention heads, which provably reduces redundant pairwise interactions while ensuring sufficient and diverse coverage. This leads to an \emph{inception-like functional diversity} in the attention heads, and promotes more informative and disentangled representations. We integrate Fibottention into standard Transformer architectures and conduct extensive experiments across multiple domains, including image classification, video understanding, and robot learning. Results demonstrate that models equipped with Fibottention either significantly outperform or achieve on-par performance with their dense MHSA counterparts, while leveraging only $2\%$ of all pairwise interactions across self-attention heads in typical settings, $2-6\%$ of the pairwise interactions in self-attention heads, resulting in substantial computational savings. Moreover, when compared to existing sparse attention mechanisms, Fibottention consistently achieves superior results on a FLOP-equivalency basis. Finally, we provide an in-depth analysis of the enhanced feature diversity resulting from our attention design and discuss its implications for efficient representation learning.

Christian Kümmerle, Claudio Mayrink Verdun

We propose an iterative algorithm for low-rank matrix completion that can be interpreted as an iteratively reweighted least squares (IRLS) algorithm, a saddle-escaping smoothing Newton method or a variable metric proximal gradient method applied to a non-convex rank surrogate. It combines the favorable data-efficiency of previous IRLS approaches with an improved scalability by several orders of magnitude. We establish the first local convergence guarantee from a minimal number of samples for that class of algorithms, showing that the method attains a local quadratic convergence rate. Furthermore, we show that the linear systems to be solved are well-conditioned even for very ill-conditioned ground truth matrices. We provide extensive experiments, indicating that unlike many state-of-the-art approaches, our method is able to complete very ill-conditioned matrices with a condition number of up to $10^{10}$ from few samples, while being competitive in its scalability.

Christian Kümmerle, Claudio Mayrink Verdun, Dominik Stöger

The recovery of sparse data is at the core of many applications in machine learning and signal processing. While such problems can be tackled using $\ell_1$-regularization as in the LASSO estimator and in the Basis Pursuit approach, specialized algorithms are typically required to solve the corresponding high-dimensional non-smooth optimization for large instances. Iteratively Reweighted Least Squares (IRLS) is a widely used algorithm for this purpose due its excellent numerical performance. However, while existing theory is able to guarantee convergence of this algorithm to the minimizer, it does not provide a global convergence rate. In this paper, we prove that a variant of IRLS converges with a global linear rate to a sparse solution, i.e., with a linear error decrease occurring immediately from any initialization, if the measurements fulfill the usual null space property assumption. We support our theory by numerical experiments showing that our linear rate captures the correct dimension dependence. We anticipate that our theoretical findings will lead to new insights for many other use cases of the IRLS algorithm, such as in low-rank matrix recovery.

Christian Kümmerle, Claudio M. Verdun

We propose an iterative algorithm for low-rank matrix completion that can be interpreted as both an iteratively reweighted least squares (IRLS) algorithm and a saddle-escaping smoothing Newton method applied to a non-convex rank surrogate objective. It combines the favorable data efficiency of previous IRLS approaches with an improved scalability by several orders of magnitude. Our method attains a local quadratic convergence rate already for a number of samples that is close to the information theoretical limit. We show in numerical experiments that unlike many state-of-the-art approaches, our approach is able to complete very ill-conditioned matrices with a condition number of up to $10^{10}$ from few samples.

Dominik Alfke, Weston Baines, Jan Blechschmidt et al.

We present a novel technique based on deep learning and set theory which yields exceptional classification and prediction results. Having access to a sufficiently large amount of labelled training data, our methodology is capable of predicting the labels of the test data almost always even if the training data is entirely unrelated to the test data. In other words, we prove in a specific setting that as long as one has access to enough data points, the quality of the data is irrelevant.