Estimating canopy height and its changes at meter resolution from satellite imagery remains a challenging computer vision task with critical environmental applications. However, the lack of open-access datasets at this resolution hinders the reproducibility and evaluation of models. We introduce Open-Canopy, the first open-access, country-scale benchmark for very high-resolution (1.5 m) canopy height estimation, covering over 87,000 km2across France with 1.5 m panchromatic resolution satellite imagery and aerial LiDAR data. Additionally, we present Open-Canopy-∆, a benchmark for canopy height reduction detection between images from different years at tree level—a difficult task for current computer vision models. We evaluate state-of-the-art architectures on these benchmarks, highlighting significant challenges and opportunities for improvement. Our datasets and code are publicly available at https://github.com/fajwel/Open-Canopy.

Significant efforts have been directed towards adapting self-supervised multimodal learning for Earth observation applications. However, most current methods produce coarse patch-sized embeddings, limiting their effectiveness and integration with other modalities like LiDAR. To close this gap, we present DUNIA, an approach to learn pixel-sized embeddings through cross-modal alignment between images and full-waveform LiDAR data. As the model is trained in a contrastive manner, the embeddings can be directly leveraged in the context of a variety of environmental monitoring tasks in a zero-shot setting. In our experiments, we demonstrate the effectiveness of the embeddings for seven such tasks: canopy height mapping, fractional canopy cover, land cover mapping, tree species identification, plant area index, crop type classification, and per-pixel waveform-based vertical structure mapping. The results show that the embeddings, along with zero-shot classifiers, often outperform specialized supervised models, even in low-data regimes. In the fine-tuning setting, we show strong performances near or better than the state-of-the-art on five out of six tasks.

The Frank-Wolfe method solves smooth constrained convex optimization problems at a generic sublinear rate of $\mathcal{O}(1/T)$, and it (or its variants) enjoys accelerated convergence rates for two fundamental classes of constraints: polytopes and strongly-convex sets. Uniformly convex sets non-trivially subsume strongly convex sets and form a large variety of \textit{curved} convex sets commonly encountered in machine learning and signal processing. For instance, the $\ell_p$-balls are uniformly convex for all $p > 1$, but strongly convex for $p\in]1,2]$ only. We show that these sets systematically induce accelerated convergence rates for the original Frank-Wolfe algorithm, which continuously interpolate between known rates. Our accelerated convergence rates emphasize that it is the curvature of the constraint sets – not just their strong convexity – that leads to accelerated convergence rates. These results also importantly highlight that the Frank-Wolfe algorithm is adaptive to much more generic constraint set structures, thus explaining faster empirical convergence. Finally, we also show accelerated convergence rates when the set is only locally uniformly convex around the optima and provide similar results in online linear optimization.

Linear bandit algorithms yield $\tilde{\mathcal{O}}(n\sqrt{T})$ pseudo-regret bounds on compact convex action sets $\mathcal{K}\subset\mathbb{R}^n$ and two types of structural assumptions lead to better pseudo-regret bounds. When $\mathcal{K}$ is the simplex or an $\ell_p$ ball with $p\in]1,2]$, there exist bandits algorithms with $\tilde{\mathcal{O}}(\sqrt{nT})$ pseudo-regret bounds. Here, we derive bandit algorithms for some strongly convex sets beyond $\ell_p$ balls that enjoy pseudo-regret bounds of $\tilde{\mathcal{O}}(\sqrt{nT})$. This result provides new elements for the open question in Bubeck and Cesa-Bianchi, 2012. When the action set is $q$-uniformly convex but not necessarily strongly convex ($q >2$), we obtain pseudo-regret bounds $\tilde{\mathcal{O}}(n^{1/q}T^{1/p})$ with $p$ s.t. $1/p + 1/q=1$. These pseudo-regret bounds are competitive with the general $\tilde{\mathcal{O}}(n\sqrt{T})$ for a time horizon range that depends on the degree $q>2$ of the set's uniform convexity and the dimension $n$ of the problem.

Given a measurement graph $G= (V,E)$ and an unknown signal $r \in \mathbb{R}^n$, we investigate algorithms for recovering $r$ from pairwise measurements of the form $r_i - r_j$; $\{i,j\} \in E$. This problem arises in a variety of applications, such as ranking teams in sports data and time synchronization of distributed networks. Framed in the context of ranking, the task is to recover the ranking of $n$ teams (induced by $r$) given a small subset of noisy pairwise rank offsets. We propose a simple SVD-based algorithmic pipeline for both the problem of time synchronization and ranking. We provide a detailed theoretical analysis in terms of robustness against both sampling sparsity and noise perturbations with outliers, using results from matrix perturbation and random matrix theory. Our theoretical findings are complemented by a detailed set of numerical experiments on both synthetic and real data, showcasing the competitiveness of our proposed algorithms with other state-of-the-art methods.

Due to its linear complexity, naive Bayes classification remains an attractive supervised learning method, especially in very large-scale settings. We propose a sparse version of naive Bayes, which can be used for feature selection. This leads to a combinatorial maximum-likelihood problem, for which we provide an exact solution in the case of binary data, or a bound in the multinomial case. We prove that our bound becomes tight as the marginal contribution of additional features decreases. Both binary and multinomial sparse models are solvable in time almost linear in problem size, representing a very small extra relative cost compared to the classical naive Bayes. Numerical experiments on text data show that the naive Bayes feature selection method is as statistically effective as state-of-the-art feature selection methods such as recursive feature elimination, l_1-penalized logistic regression and LASSO, while being orders of magnitude faster. For a large data set, having more than with 1.6 million training points and about 12 million features, and with a non-optimized CPU implementation, our sparse naive Bayes model can be trained in less than 15 seconds.

We design simple screening tests to automatically discard data samples in empirical risk minimization withoutlosing optimization guarantees. We derive loss functions that produce dual objectives with a sparse solution. We also show how to regularize convex losses to ensure such a dual sparsity-inducing property, andpropose a general method to design screening tests for classification or regression based on ellipsoidal approximations of the optimal set. In addition to producing computational gains, our approach also allows us to compress a dataset into a subset of representative points.

Estimating Wasserstein distances between two high-dimensional densities suffers from the curse of dimensionality: one needs an exponential (wrt dimension) number of samples to ensure that the distance between two empirical measures is comparable to the distance between the original densities. Therefore, optimal transport (OT) can only be used in machine learning if it is substantially regularized. On the other hand, one of the greatest achievements of the OT literature in recent years lies in regularity theory: Caffarelli showed that the OT map between two well behaved measures is Lipschitz, or equivalently when considering 2-Wasserstein distances, that Brenier convex potentials (whose gradient yields an optimal map) are smooth. We propose in this work to draw inspiration from this theory and use regularity as a regularization tool. We give algorithms operating on two discrete measures that can recover nearly optimal transport maps with small distortion, or equivalently, nearly optimal Brenier potentials that are strongly convex and smooth. The problem boils down to solving alternatively a convex QCQP and a discrete OT problem, granting access to the values and gradients of the Brenier potential not only on sampled points, but also out of sample at the cost of solving a simpler QCQP for each evaluation. We propose algorithms to estimate and evaluate transport maps with desired regularity properties, benchmark their statistical performance, apply them to domain adaptation and visualize their action on a color transfer task.

In smooth strongly convex optimization, knowledge of the strong convexity parameter is critical for obtaining simple methods with accelerated rates. In this work, we study a class of methods, based on Polyak steps, where this knowledge is substituted by that of the optimal value, $f_*$. We first show slightly improved convergence bounds than previously known for the classical case of simple gradient descent with Polyak steps, we then derive an accelerated gradient method with Polyak steps and momentum, along with convergence guarantees.

We describe a convergence acceleration scheme for multi-step optimization algorithms. The extrapolated solution is written as a nonlinear average of the iterates produced by the original optimization algorithm. Our scheme does not need the underlying fixed-point operator to be symmetric, hence handles e.g. algorithms with momentum terms such as Nesterov’s accelerated method, or primal-dual methods such as Chambolle-Pock. The weights are computed via a simple linear system and we analyze performance in both online and offline modes. We use Crouzeix’s conjecture to show that acceleration is controlled by the solution of a Chebyshev problem on the numerical range of a nonsymmetric operator modelling the behavior of iterates near the optimum. Numerical experiments are detailed on image processing and logistic regression problems.

Conditional Gradients (aka Frank-Wolfe algorithms) form a classical set of methods for constrained smooth convex minimization due to their simplicity, the absence of projection step, and competitive numerical performance. While the vanilla Frank-Wolfe algorithm only ensures a worst-case rate of $O(1/\epsilon)$, various recent results have shown that for strongly convex functions, the method can be slightly modified to achieve linear convergence. However, this still leaves a huge gap between sublinear $O(1/\epsilon)$ convergence and linear $O(\log 1/\epsilon)$ convergence to reach an $\epsilon$-approximate solution. Here, we present a new variant of Conditional Gradients, that can dynamically adapt to the function’s geometric properties using restarts and thus smoothly interpolates between the sublinear and linear regimes. Furthermore, our results apply to generic compact convex constraint sets.

We present a novel algorithm for overcomplete independent components analysis (ICA), where the number of latent sources k exceeds the dimension p of observed variables. Previous algorithms either suffer from high computational complexity or make strong assumptions about the form of the mixing matrix. Our algorithm does not make any sparsity assumption yet enjoys favorable computational and theoretical properties. Our algorithm consists of two main steps: (a) estimation of the Hessians of the cumulant generating function (as opposed to the fourth and higher order cumulants used by most algorithms) and (b) a novel semi-definite programming (SDP) relaxation for recovering a mixing component. We show that this relaxation can be efficiently solved with a projected accelerated gradient descent method, which makes the whole algorithm computationally practical. Moreover, we conjecture that the proposed program recovers a mixing component at the rate $k < p^2/4$ and prove that a mixing component can be recovered with high probability when $k <(2 - \epsilon)p\log p$ when the original components are sampled uniformly at random on the hyper sphere. Experiments are provided on synthetic data and the CIFAR-10 dataset of real images.

We analyze two novel randomized variants of the Frank-Wolfe (FW) or conditional gradient algorithm. While classical FW algorithms require solving a linear minimization problem over the domain at each iteration, the proposed method only requires to solve a linear minimization problem over a small subset of the original domain. The first algorithm that we propose is a randomized variant of the original FW algorithm and achieves a $\mathcal{O}(1/t)$ sublinear convergence rate as in the deterministic counterpart. The second algorithm is a randomized variant of the Away-step FW algorithm, and again as its deterministic counterpart, reaches linear (i.e., exponential) convergence rate making it the first provably convergent randomized variant of Away-step FW. In both cases, while subsampling reduces the convergence rate by a constant factor, the linear minimization step can be a fraction of the cost of that of the deterministic versions, especially when the data is streamed. We illustrate computational gains of both algorithms on regression problems, involving both $\ell_1$ and latent group lasso penalties.

The {\L}ojasiewicz inequality shows that H\"olderian error bounds on the minimum of convex optimization problems hold almost generically. Here, we clarify results of \citet{Nemi85} who show that H\"olderian error bounds directly controls the performance of restart schemes. The constants quantifying error bounds are of course unobservable, but we show that optimal restart strategies are robust, and searching for the best scheme only increases the complexity by a logarithmic factor compared to the optimal bound. Overall then, restart schemes generically accelerate accelerated methods.

Extrapolation methods use the last few iterates of an optimization algorithm to produce a better estimate of the optimum. They were shown to achieve optimal convergence rates in a deterministic setting using simple gradient iterates. Here, we study extrapolation methods in a stochastic setting, where the iterates are produced by either a simple or an accelerated stochastic gradient algorithm. We first derive convergence bounds for arbitrary, potentially biased perturbations, then produce asymptotic bounds using the ratio between the variance of the noise and the accuracy of the current point. Finally, we apply this acceleration technique to stochastic algorithms such as SGD, SAGA, SVRG and Katyusha in different settings, and show significant performance gains.

We show that accelerated optimization methods can be seen as particular instances of multi-step integration schemes from numerical analysis, applied to the gradient flow equation. Compared with recent advances in this vein, the differential equation considered here is the basic gradient flow, and we derive a class of multi-step schemes which includes accelerated algorithms, using classical conditions from numerical analysis. Multi-step schemes integrate the differential equation using larger step sizes, which intuitively explains the acceleration phenomenon.

Abstract Motivation New long read sequencers promise to transform sequencing and genome assembly by producing reads tens of kilobases long. However, their high error rate significantly complicates assembly and requires expensive correction steps to layout the reads using standard assembly engines. Results We present an original and efficient spectral algorithm to layout the uncorrected nanopore reads, and its seamless integration into a straightforward overlap/layout/consensus (OLC) assembly scheme. The method is shown to assemble Oxford Nanopore reads from several bacterial genomes into good quality (∼99% identity to the reference) genome-sized contigs, while yielding more fragmented assemblies from the eukaryotic microbe Sacharomyces cerevisiae. Availability and implementation https://github.com/antrec/spectrassembler. Supplementary Information Supplementary data are available at Bioinformatics online.

We describe a convergence acceleration technique for generic optimization problems. Our scheme computes estimates of the optimum from a nonlinear average of the iterates produced by any optimization method. The weights in this average are computed via a simple and small linear system, whose solution can be updated online. This acceleration scheme runs in parallel to the base algorithm, providing improved estimates of the solution on the fly, while the original optimization method is running. Numerical experiments are detailed on classical classification problems.

We describe a seriation algorithm for ranking a set of items given pairwise comparisons between these items. Intuitively, the algorithm assigns similar rankings to items that compare similarly with all others. It does so by constructing a similarity matrix from pairwise comparisons, using seriation methods to reorder this matrix and construct a ranking. We first show that this spectral seriation algorithm recovers the true ranking when all pairwise comparisons are observed and consistent with a total order. We then show that ranking reconstruction is still exact when some pairwise comparisons are corrupted or missing, and that seriation based spectral ranking is more robust to noise than classical scoring methods. Finally, we bound the ranking error when only a random subset of the comparions are observed. An additional benefit of the seriation formulation is that it allows us to solve semi-supervised ranking problems. Experiments on both synthetic and real datasets demonstrate that seriation based spectral ranking achieves competitive and in some cases superior performance compared to classical ranking methods.

Starting from a multivariate data set, we study several techniques to isolate affine combinations of the variables with a maximum amount of mean reversion, while constraining the variance to be larger than a given threshold. We show that many of the optimization problems arising in this context can be solved exactly using semidefinite programming and some variant of the \mathcalS-lemma. In finance, these methods are used to isolate statistical arbitrage opportunities, i.e. mean reverting portfolios with enough variance to overcome market friction. In a more general setting, mean reversion and its generalizations are also used as a proxy for stationarity, while variance simply measures signal strength.

Seriation seeks to reconstruct a linear order between variables using unsorted similarity information. It has direct applications in archeology and shotgun gene sequencing for example. We prove the equivalence between the seriation and the combinatorial 2-sum problem (a quadratic minimization problem over permutations) over a class of similarity matrices. The seriation problem can be solved exactly by a spectral algorithm in the noiseless case and we produce a convex relaxation for the 2-sum problem to improve the robustness of solutions in a noisy setting. This relaxation also allows us to impose additional structural constraints on the solution, to solve semi-supervised seriation problems. We present numerical experiments on archeological data, Markov chains and gene sequences.

We propose new methodologies to detect anomalies in discrete-time processes taking values in a set. The method is based on the inference of functionals whose evaluations on successive states visited by the process have low autocorrelations. Deviations from this behavior are used to flag anomalies. The candidate functionals are estimated in a subset of a reproducing kernel Hilbert space associated with the set where the process takes values. We provide experimental results which show that these techniques compare favorably with other algorithms.

Given a sample covariance matrix, we examine the problem of maximizing the variance explained by a particular linear combination of the input variables while constraining the number of nonzero coefficients in this combination. This is known as sparse principal component analysis and has a wide array of applications in machine learning and engineering. We formulate a new semidefinite relaxation to this problem and derive a greedy algorithm that computes a full set of good solutions for all numbers of non zero coefficients, with complexity O(n3), where n is the number of variables. We then use the same relaxation to derive sufficient conditions for global optimality of a solution, which can be tested in O(n3). We show on toy examples and biological data that our algorithm does provide globally optimal solutions in many cases.

In this paper, we propose a method for support vector machine classification using indefinite kernels. Instead of directly minimizing or stabilizing a nonconvex loss function, our method simultaneously finds the support vectors and a proxy kernel matrix used in computing the loss. This can be interpreted as a robust classification problem where the indefinite kernel matrix is treated as a noisy observation of the true positive semidefinite kernel. Our formulation keeps the problem convex and relatively large problems can be solved efficiently using the analytic center cutting plane method. We compare the performance of our technique with other methods on several data sets.

We consider the problem of fitting a large-scale covariance matrix to multivariate Gaussian data in such a way that the inverse is sparse, thus providing model selection. Beginning with a dense empirical covariance matrix, we solve a maximum likelihood problem with an l1-norm penalty term added to encourage sparsity in the inverse. For models with tens of nodes, the resulting problem can be solved using standard interior-point algorithms for convex optimization, but these methods scale poorly with problem size. We present two new algorithms aimed at solving problems with a thousand nodes. The first, based on Nesterov's first-order algorithm, yields a rigorous complexity estimate for the problem, with a much better dependence on problem size than interior-point methods. Our second algorithm uses block coordinate descent, updating row/columns of the covariance matrix sequentially. Experiments with genomic data show that our method is able to uncover biologically interpretable connections among genes.

We examine the problem of approximating, in the Frobenius-norm sense, a positive, semidefinite symmetric matrix by a rank-one matrix, with an upper bound on the cardinality of its eigenvector. The problem arises in the decomposition of a covariance matrix into sparse factors, and has wide applications ranging from biology to finance. We use a modifica- tion of the classical variational representation of the largest eigenvalue of a symmetric matrix, where cardinality is constrained, and derive a semidefinite programming based relaxation for our problem.