Transformers can learn noisy linear systems with depth and IID data.
problem Learning noisy linear dynamical systems with transformers.
method Theoretical analysis of multi-layer and single-layer transformers with respect to L 2 L^2 L 2 -testing loss. result Single-layer transformers have a non-diminishing lower bound on approximation error, suggesting depth separation.
Deep autoencoder finds linear PDE coordinates for nonlinear equations.
problem Discovering linear coordinates for nonlinear PDEs.
method Residual network architecture for finding intrinsic coordinates.
result Deep learning autoencoder transforms nonlinear PDEs into linear ones.
Transformers become faster by linearizing self-attention.
problem Quadratic complexity of transformers makes them slow for long sequences.
method Expressed self-attention as a linear dot-product and used matrix product associativity to reduce complexity.
result Linear transformers are up to 4000x faster on long sequences.
Paper proposes linear transformers for efficient in-context learning without context length limitations.
problem Quadratic complexity of softmax transformers limits data processing speed.
method Investigates linear transformers under domain generalization, showing they learn mappings from context distributions to response functions.
result Linear transformers achieve in-context learning with a linear complexity in context length, offering a dimension-independent convergence rate.
Geometrically transforms nonconservative dynamics to linearize Kepler and Manev systems.
problem Regularizing and linearizing nonconservative central force dynamics.
method Projective transformation and conformal scaling in configuration and phase spaces.
result Full linearization of Kepler and Manev dynamics in any finite dimension.
Linear attention in Transformers can be interpreted as dynamic VAR models.
problem Misalignment between Transformers and autoregressive forecasting objectives.
method Interpreting linear attention as VAR, rearranging MLP, attention, and flow.
result SAMoVAR improves performance, interpretability, and efficiency.
Proposes σ σ σ -PCA to learn identifiable linear transformations without whitening.
problem Cannot identify axes with equal variances in PCA.
method Unified model for linear and nonlinear PCA, introducing a missing piece to eliminate rotational indeterminacy.
result Eliminates subspace rotational indeterminacy in PCA.
New linear flows using exponential of linear transformations improve generative models.
problem Improving generative models in machine learning.
method Developed convolution exponentials and generalized Sylvester Flows using the exponential of linear transformations.
result Convolution exponentials and Convolutional Sylvester Flows outperform other models in log-likelihood.
GLU variants improve Transformer performance.
problem Improving the performance of Transformer models.
method Tested variations of Gated Linear Units (GLU) in Transformer feed-forward sublayers.
result Some GLU variants yield better performance than ReLU or GELU activations.
OLS is a special case of Transformer, revealing its linear nature.
problem Understanding the statistical essence of Transformer architecture.
method Algebraic proof and spectral decomposition of covariance matrix.
result Attention mechanism in Transformers is mathematically equivalent to OLS.
Transformers can approximate Newton's method for logistic regression.
problem Implementing higher order optimization methods in Transformers.
method Linear attention Transformers with ReLU layers approximating second order optimization algorithms.
result Transformers can implement a single step of Newton's iteration for matrix inversion.
Calculates affine transformations for specific homogeneous spaces.
problem Computing groups of affine transformations on homogeneous spaces.
method Analyzes conditions for affine connections and uses them to establish group isomorphisms.
result Groups of affine transformations are locally isomorphic under specified conditions.
B-cos transformers explain Vision Transformers' decisions.
problem Lack of holistic explanations for transformer outputs.
method Formulate each component as dynamic linear, allowing a single linear transform for summarization.
result Bcos-ViTs are highly interpretable and competitive on ImageNet.
Transformers can handle endogeneity in linear regression using IV methods.
problem Endogeneity in in-context linear regression models.
method Transformer architecture with gradient-based bi-level optimization and in-context pretraining.
result Transformers provide more robust predictions and estimates than 2SLS in endogenous scenarios.
Transformers learn linear models in-context without updates.
problem Understanding how transformers mimic linear models in-context.
method Gradient flow on linear regression tasks with random initialization.
result Transformers achieve prediction error competitive with best linear predictors.
Transformers trained on random classification tasks generalize well and can overfit without error.
problem Understanding how transformers generalize and overfit in-context.
method Analysis of implicit regularization during gradient descent training.
result Transformers can overfit without error and still generalize well.
Study linear transformations' effects on data augmentation for improved estimation.
problem Improving performance in image and text classification tasks.
method Examined a family of linear transformations in over-parametrized linear regression settings.
result Transformations that preserve labels or mix data can improve estimation.
Linformer reduces transformer complexity to linear, improving efficiency.
problem High cost of training and deploying large transformer models for long sequences.
method Approximates self-attention with low-rank matrix, proposing Linformer with O ( n ) O(n) O ( n ) complexity. result Linformer performs similarly to standard transformers but is more memory- and time-efficient.
New method linearizes Darboux transformations of discrete curves.
problem Linearizing Darboux transformations of discrete curves.
method Expressing Darboux transformations as parallel sections of discrete connections in quaternionic formalism.
result Closed-form discrete parametrisations of all Darboux transforms and bicycle correspondences.
Efficient algorithms find optimal monotone transforms for calibration under strictly convex losses.
problem Calibrating estimations to improve performance with monotone transforms.
method Proposed linear-time and space algorithm for finding optimal monotone transforms for specific loss functions. Also proposed an anytime algorithm with linear space and pseudo-linearithmic time complexity.
result Optimal monotone transforms are unique and can be found efficiently for various strictly convex loss functions.
Transformers can be hijacked by context, but deeper models are more robust.
problem Robustness of Transformers against context hijacking for linear classification.
method Developed a theoretical analysis on the robustness of linear transformers, considering model depth, training context lengths, and number of hijacking context tokens.
result Deeper transformers are more robust to context hijacking.
Paper introduces a new method for Transformers with linear complexity.
problem No efficient relative positional encoding for linear Transformer models.
method Stochastic Positional Encoding (SPE) that replaces classical RPE.
result SPE behaves like RPE and performs well on benchmarks.
Deep linear ResNets converge globally with certain transformations.
problem Global convergence of training deep linear ResNets.
method Gradient descent and stochastic gradient descent for training L L L -hidden-layer linear ResNets. result GD and SGD can converge to global minimum for deep linear ResNets with specific transformations.
Legendre transformations link related integrable hierarchies.
problem Understanding relationships between integrable hierarchies.
method Legendre-type transformations of generalized Frobenius manifolds.
result Linear reciprocal transformations link related hierarchies.
LLT transforms time series features based on linear laws.
problem Classifying univariate and multivariate time series.
method Time-delay embedding, spectral decomposition, and feature transformation.
result Transformed features improve classification accuracy.
Paper recovers latent causal structure and linear transformation from indirect observations.
problem Recovering latent causal structure and linear transformation from indirect observations.
method Established sufficient conditions for DAG recovery, leveraged score function properties, and used soft/hard interventions.
result Perfect recovery of latent DAG structure and linear transformation up to scaling using soft interventions, hard interventions with additional hypothesis testing.
We show how to convert ICL in linearized transformers into model weights.
problem Making in-context learning interpretable and permanent in large language models.
method Demonstrates equivalence between ICL and bias terms in linearized transformers, and develops ICLCA for exact conversion.
result Exact conversion of in-context learning into model weights is possible for linearized transformers.
B-cos transforms improve neural network interpretability by aligning weights.
problem Improving interpretability of deep neural networks.
method Replacing linear transforms with B-cos transforms that promote weight-input alignment during training.
result B-cos transforms lead to highly interpretable and task-relevant linear summaries of neural network computations.
State spaces of multifactor approximations of nonnegative Volterra processes are linear transformations of the nonnegative orthant.
problem Characterizing state spaces of multifactor approximations of nonnegative Volterra processes.
method Explicit linear transformation of the nonnegative orthant.
result State spaces of multifactor approximations of nonnegative Volterra processes are given by explicit linear transformation of the nonnegative orthant.
Kernel models learn low-dimensional predictive subspaces from input data.
problem Learning effective feature transformations in kernel models.
method Study of a compositional kernel ridge regression model.
result Global minimizers of the objective function identify the subspace with high probability.
Our goal is to extract meaningful transformations from raw images, such as varying the thickness of lines in handwriting or the lighting in a portrait. We propose an unsupervised approach to learn such transformations by attempting to reconstruct an image from a linear combination of transformations of its nearest neig…
Injectivity of geodesic ray transform on specific Finsler manifolds proven.
problem Injectivity of geodesic ray transform on spherically symmetric reversible Finsler manifolds.
method Reduction to invertibility of generalized Abel transforms using angular Fourier series and Taylor expansions of geodesics.
result Injectivity of geodesic ray transform proven on specified Finsler manifolds.
Study on neural scaling laws for solving linear systems in-context.
problem Theoretical guarantees for solving linear systems using a linear transformer architecture.
method Neural scaling laws and task diversity for in-domain and out-of-domain generalization.
result Novel notion of task diversity for necessary and sufficient condition of generalization under task shifts.
In order to understand the linearization problem around a leaf of a singular foliation, we extend the familiar holonomy map from the case of regular foliations to the case of singular foliations. To this aim we introduce the notion of holonomy transformation. Unlike the regular case, holonomy transformations can not be…
Transitive consistency is an intrinsic property for collections of linear invertible transformations between Euclidean coordinate frames. In practice, when the transformations are estimated from data, this property is lacking. This work addresses the problem of synchronizing transformations that are not transitively co…
This work solves TRPCA under linear transforms, recovering low-rank and sparse components.
problem Exact recovery of tensor low-rank and sparse components from their sum.
method Convex optimization with weighted tensor nuclear norm and ℓ1-norm.
result The convex program exactly recovers the components under certain incoherence conditions.
Linear Transformer Block combines MLP and linear attention for near-optimal ICL in linear regression.
problem Achieving near-optimal in-context learning (ICL) risk for linear regression with a Gaussian prior.
method Combines linear attention and MLP components in a Linear Transformer Block (LTB). Establishes correspondence with one-step gradient descent estimators ( G D e x t − β \mathsf{GD} ext{-}\mathbfβ GD e x t − β ). result LTB achieves nearly Bayes optimal ICL risk for linear regression with a Gaussian prior.
This paper uses linear rational splines for invertible modeling, offering a simpler inverse and similar costs.
problem Creating expressive invertible models with tractable Jacobian determinants.
method Replacing affine transformations with linear rational splines in coupling layers.
result Linear rational splines offer a simpler inverse and similar costs for inference and generation.
Examines how first-order differential operators can be equivalently transformed.
problem Equivalence of first-order linear differential operators.
method Discussion of equivalency transformations.
result Explains how first-order differential operators can be transformed equivalently.
Transformers can learn mixture of linear models efficiently.
problem Existence and generalization of in-context learning for mixture models.
method Theoretical analysis and gradient flow optimization.
result Transformers achieve a prediction error of O ( d / n ) \mathcal{O}(\sqrt{d/n}) O ( d / n ) with high probability. New insights into how large learning rates affect transformer training dynamics.
problem Understanding how large learning rates impact the training of transformer models.
method Analyzing a simplified linear transformer model with a two-factor product map.
result Large learning rates can lead to various training outcomes including cycles, chaos, or divergence.
NDMs enable non-linear transformations in diffusion models for better generative tasks.
problem Limited to linear transformations, diffusion models struggle with generative tasks.
method Presented NDMs that allow time-dependent non-linear transformations.
result NDMs outperform conventional diffusion models in likelihood and sample quality.
Transformers can approximate Kalman Filtering in linear systems with small error.
problem Approximating Kalman Filtering using Transformers for linear dynamical systems.
method Two-step reduction: 1) Softmax self-attention block approximates Nadaraya-Watson kernel smoothing, 2) This estimator approximates Kalman Filter.
result Constructs a Transformer that implements the Kalman Filter with small additive error, uniformly bounded in time.
We present a signal representation framework called the sparse manifold transform that combines key ideas from sparse coding, manifold learning, and slow feature analysis. It turns non-linear transformations in the primary sensory signal space into linear interpolations in a representational embedding space while maint…
This work introduces a transformation-based learner model for classification forests. The weak learner at each split node plays a crucial role in a classification tree. We propose to optimize the splitting objective by learning a linear transformation on subspaces using nuclear norm as the optimization criteria. The le…
This paper compares Transformers and RNNs in various tasks, showing size differences.
problem Comparing representational capabilities of Transformers and RNNs across tasks.
method Analysis of differences in tasks like index lookup, nearest neighbor, and string equality.
result Size differences in Transformers and RNNs for various tasks.
Improved singular value approximation for convolutional layers.
problem Improving accuracy of singular value approximation for linear convolutional layers.
method Developed a new spectral density matrix method for singular value approximation with improved accuracy and reduced computational complexity.
result Obtained moderate improvement in singular value distribution compared to circular approximation.
ZeroS improves Transformers by adding negative weights, matching or beating softmax attention.
problem Limited performance of linear attention methods, especially in long context sequences.
method Proposes Zero-Sum Linear Attention (ZeroS) that removes the zero-order term and reweights zero-sum softmax residuals.
result ZeroS matches or exceeds standard softmax attention across various benchmarks, theoretically expanding representable functions.