arXiv subjects

Allan Grønlund

Publications and source records attributed to Allan Grønlund.

At least 19 recordsLinked to original sources

arXiv2025

An Exponential Separation Between Quantum and Quantum-Inspired Classical Algorithms for Linear Systems

Achieving a provable exponential quantum speedup for an important machine learning task has been a central research goal since the seminal HHL quantum algorithm for solving linear systems and the subsequent quantum recommender systems algorithm by Kerenidis and Prakash. These algorithms were initially believed to be strong candidates for exponential speedups, but a lower bound ruling out similar classical improvements remained absent. In breakthrough work by Tang, it was demonstrated that this lack of progress in classical lower bounds was for good reasons. Concretely, she gave a classical counterpart of the quantum recommender systems algorithm, reducing the quantum advantage to a mere polynomial. Her approach is quite general and was named quantum-inspired classical algorithms. Since then, almost all the initially exponential quantum machine learning speedups have been reduced to polynomial via new quantum-inspired classical algorithms. From the current state-of-affairs, it is unclear whether we can hope for exponential quantum speedups for any natural machine learning task. In this work, we present the first such provable exponential separation between quantum and quantum-inspired classical algorithms for the basic problem of solving a linear system when the input matrix is well-conditioned and has sparse rows and columns.

quant-ph↗

arXiv2024

The NFA Acceptance Hypothesis: Non-Combinatorial and Dynamic Lower Bounds

We pose the fine-grained hardness hypothesis that the textbook algorithm for the NFA Acceptance problem is optimal up to subpolynomial factors, even for dense NFAs and fixed alphabets. We show that this barrier appears in many variations throughout the algorithmic literature by introducing a framework of Colored Walk problems. These yield fine-grained equivalent formulations of the NFA Acceptance problem as problems concerning detection of an $s$-$t$-walk with a prescribed color sequence in a given edge- or node-colored graph. For NFA Acceptance on sparse NFAs (or equivalently, Colored Walk in sparse graphs), a tight lower bound under the Strong Exponential Time Hypothesis has been rediscovered several times in recent years. We show that our hardness hypothesis, which concerns dense NFAs, has several interesting implications: - It gives a tight lower bound for Context-Free Language Reachability. This proves conditional optimality for the class of 2NPDA-complete problems, explaining the cubic bottleneck of interprocedural program analysis. - It gives a tight $(n+nm^{1/3})^{1-o(1)}$ lower bound for the Word Break problem on strings of length $n$ and dictionaries of total size $m$. - It implies the popular OMv hypothesis. Since the NFA acceptance problem is a static (i.e., non-dynamic) problem, this provides a static reason for the hardness of many dynamic problems. Thus, a proof of the NFA Acceptance hypothesis would resolve several interesting barriers. Conversely, a refutation of the NFA Acceptance hypothesis may lead the way to attacking the current barriers observed for Context-Free Language Reachability, the Word Break problem and the growing list of dynamic problems proven hard under the OMv hypothesis.

cs.CC↗

arXiv2023

Sublinear Time Shortest Path in Expander Graphs

Computing a shortest path between two nodes in an undirected unweighted graph is among the most basic algorithmic tasks. Breadth first search solves this problem in linear time, which is clearly also a lower bound in the worst case. However, several works have shown how to solve this problem in sublinear time in expectation when the input graph is drawn from one of several classes of random graphs. In this work, we extend these results by giving sublinear time shortest path (and short path) algorithms for expander graphs. We thus identify a natural deterministic property of a graph (that is satisfied by typical random regular graphs) which suffices for sublinear time shortest paths. The algorithms are very simple, involving only bidirectional breadth first search and short random walks. We also complement our new algorithms by near-matching lower bounds.

cs.DS↗

arXiv2021

Compression Implies Generalization

Explaining the surprising generalization performance of deep neural networks is an active and important line of research in theoretical machine learning. Influential work by Arora et al. (ICML'18) showed that, noise stability properties of deep nets occurring in practice can be used to provably compress model representations. They then argued that the small representations of compressed networks imply good generalization performance albeit only of the compressed nets. Extending their compression framework to yield generalization bounds for the original uncompressed networks remains elusive. Our main contribution is the establishment of a compression-based framework for proving generalization bounds. The framework is simple and powerful enough to extend the generalization bounds by Arora et al. to also hold for the original network. To demonstrate the flexibility of the framework, we also show that it allows us to give simple proofs of the strongest known generalization bounds for other popular machine learning models, namely Support Vector Machines and Boosting.

cs.LG↗

arXiv2021

Learning to Detect Fortified Areas

High resolution data models like grid terrain models made from LiDAR data are a prerequisite for modern day Geographic Information Systems applications. Besides providing the foundation for the very accurate digital terrain models, LiDAR data is also extensively used to classify which parts of the considered surface comprise relevant elements like water, buildings and vegetation. In this paper we consider the problem of classifying which areas of a given surface are fortified by for instance, roads, sidewalks, parking spaces, paved driveways and terraces. We consider using LiDAR data and orthophotos, combined and alone, to show how well the modern machine learning algorithms Gradient Boosted Trees and Convolutional Neural Networks are able to detect fortified areas on large real world data. The LiDAR data features, in particular the intensity feature that measures the signal strength of the return, that we consider in this project are heavily dependent on the actual LiDAR sensor that made the measurement. This is highly problematic, in particular for the generalisation capability of pattern matching algorithms, as this means that data features for test data may be very different from the data the model is trained on. We propose an algorithmic solution to this problem by designing a neural net embedding architecture that transforms data from all the different sensor systems into a new common representation that works as well as if the training data and test data originated from the same sensor. The final algorithm result has an accuracy above 96 percent, and an AUC score above 0.99.

cs.CV↗

arXiv2020

Margins are Insufficient for Explaining Gradient Boosting

Boosting is one of the most successful ideas in machine learning, achieving great practical performance with little fine-tuning. The success of boosted classifiers is most often attributed to improvements in margins. The focus on margin explanations was pioneered in the seminal work by Schapire et al. (1998) and has culminated in the $k$'th margin generalization bound by Gao and Zhou (2013), which was recently proved to be near-tight for some data distributions (Gronlund et al. 2019). In this work, we first demonstrate that the $k$'th margin bound is inadequate in explaining the performance of state-of-the-art gradient boosters. We then explain the short comings of the $k$'th margin bound and prove a stronger and more refined margin-based generalization bound for boosted classifiers that indeed succeeds in explaining the performance of modern gradient boosters. Finally, we improve upon the recent generalization lower bound by Grønlund et al. (2019).

cs.LG↗

arXiv2020

Near-Tight Margin-Based Generalization Bounds for Support Vector Machines

Support Vector Machines (SVMs) are among the most fundamental tools for binary classification. In its simplest formulation, an SVM produces a hyperplane separating two classes of data using the largest possible margin to the data. The focus on maximizing the margin has been well motivated through numerous generalization bounds. In this paper, we revisit and improve the classic generalization bounds in terms of margins. Furthermore, we complement our new generalization bound by a nearly matching lower bound, thus almost settling the generalization performance of SVMs in terms of margins.

cs.LG↗

arXiv2020

Margin-Based Generalization Lower Bounds for Boosted Classifiers

Boosting is one of the most successful ideas in machine learning. The most well-accepted explanations for the low generalization error of boosting algorithms such as AdaBoost stem from margin theory. The study of margins in the context of boosting algorithms was initiated by Schapire, Freund, Bartlett and Lee (1998) and has inspired numerous boosting algorithms and generalization bounds. To date, the strongest known generalization (upper bound) is the $k$th margin bound of Gao and Zhou (2013). Despite the numerous generalization upper bounds that have been proved over the last two decades, nothing is known about the tightness of these bounds. In this paper, we give the first margin-based lower bounds on the generalization error of boosted classifiers. Our lower bounds nearly match the $k$th margin bound and thus almost settle the generalization performance of boosted classifiers in terms of margins.

cs.LG↗

arXiv2020

Explaining the poor performance of the KASS algorithm implementation

By investigating the code for the KASS algorithm implementation used in the paper "Exploring the quantum speed limit with computer games" [1, arXiv:1506.09091] by Sørensen et al. (provided by the authors), we describe how the poor performance of the KASS algorithm reported in [1] is entirely caused by a simple sign error in a derivative calculation. Changing only this one sign in the KASS implementation, we show that the algorithm provides results comparable to all other algorithms considered for the problem [2,7], and performs better than all player solutions of [1]. Furthermore, we show that the player solutions were optimized with a different algorithm before being compared to the results from the KASS algorithm. The authors of [1] have acknowledged both findings. Finally, we show that in contrast to the claims in [1], the players did not explore two different strategies. In fact, all the players followed the same strategy.

math.OC↗

arXiv2019

Learning to Find Hydrological Corrections

High resolution Digital Elevation models, such as the (Big) grid terrain model of Denmark with more than 200 billion measurements, is a basic requirement for water flow modelling and flood risk analysis. However, a large number of modifications often need to be made to even very accurate terrain models, such as the Danish model, before they can be used in realistic flow modeling. These modifications include removal of bridges, which otherwise will act as dams in flow modeling, and inclusion of culverts that transport water underneath roads. In fact, the danish model is accompanied by a detailed set of hydrological corrections for the digital elevation model. However, producing these hydrological corrections is a very slow an expensive process, since it is to a large extent done manually and often with local input. This also means that corrections can be of varying quality. In this paper we propose a new algorithmic apporach based on machine learning and convolutional neural networks for automatically detecting hydrological corrections for such large terrain data. Our model is able to detect most hydrological corrections known for the danish model and quite a few more that should have been included in the original list.

cs.CV↗

arXiv2019

Algorithms Clearly Beat Gamers at Quantum Moves. A Verification

The paper [Sørensen et al., Nature 532] considers how human players compare to algorithms for solving the Quantum Moves game BringHomeWater and design new algorithms based on the intuition extracted from players. The claim by [Sørensen et al., Nature 532] is that players outperform widely used algorithms, in particular the KASS algorithm, based on the Krotov algorithm, and that player intuition is crucial to develop improved methods. However, as initially discussed by D. Sels [D. Sels, Phys. Rev. A 97], a standard Coordinate Ascent algorithm outperforms all players by a large margin. Albeit D. Sels only compare to player solutions, the simple algorithm outperforms all algorithms based on player solutions and Krotov, and it does so using much less time and iterations. In this paper we elaborate on the methods discussed by D. Sels and verify that the presented algorithm, solves the problem better than all players and algorithms derived from player solutions in [Sørensen et al., Nature 532]. We also verify the theoretical analysis presented by D. Sels, that gives a theoretically derived protocol that outperforms all players. We add a comparison with gradient ascent or GRAPE. Starting from uniform random values, GRAPE outperforms all players by a large margin. GRAPE works at least as well as the methods from [Sørensen et al., Nature 532] initialized with player solutions. A standard analysis of the results from GRAPE provides a starting point for GRAPE, that outperform all algorithms from [Sørensen et al., Nature 532]. We compare with a basic Krotov algorithm, and get results similar to GRAPE, clearly outperforming players and the KASS algorithm. These experiments verify and underline the result in [D. Sels, Phys. Rev. A 97] that the conclusions from [Sørensen et al., Nature 532] regarding algorithms are untenable. In fact the opposite conclusions are true.

cs.OH↗

arXiv2019

Optimal Minimal Margin Maximization with Boosting

Boosting algorithms produce a classifier by iteratively combining base hypotheses. It has been observed experimentally that the generalization error keeps improving even after achieving zero training error. One popular explanation attributes this to improvements in margins. A common goal in a long line of research, is to maximize the smallest margin using as few base hypotheses as possible, culminating with the AdaBoostV algorithm by (R{ä}tsch and Warmuth [JMLR'04]). The AdaBoostV algorithm was later conjectured to yield an optimal trade-off between number of hypotheses trained and the minimal margin over all training points (Nie et al. [JMLR'13]). Our main contribution is a new algorithm refuting this conjecture. Furthermore, we prove a lower bound which implies that our new algorithm is optimal.

cs.LG↗

arXiv2018

Fast Exact k-Means, k-Medians and Bregman Divergence Clustering in 1D

The $k$-Means clustering problem on $n$ points is NP-Hard for any dimension $d\ge 2$, however, for the 1D case there exists exact polynomial time algorithms. Previous literature reported an $O(kn^2)$ time dynamic programming algorithm that uses $O(kn)$ space. It turns out that the problem has been considered under a different name more than twenty years ago. We present all the existing work that had been overlooked and compare the various solutions theoretically. Moreover, we show how to reduce the space usage for some of them, as well as generalize them to data structures that can quickly report an optimal $k$-Means clustering for any $k$. Finally we also generalize all the algorithms to work for the absolute distance and to work for any Bregman Divergence. We complement our theoretical contributions by experiments that compare the practical performance of the various algorithms.

cs.DS↗

arXiv2018

Upper and lower bounds for dynamic data structures on strings

We consider a range of simply stated dynamic data structure problems on strings. An update changes one symbol in the input and a query asks us to compute some function of the pattern of length $m$ and a substring of a longer text. We give both conditional and unconditional lower bounds for variants of exact matching with wildcards, inner product, and Hamming distance computation via a sequence of reductions. As an example, we show that there does not exist an $O(m^{1/2-\varepsilon})$ time algorithm for a large range of these problems unless the online Boolean matrix-vector multiplication conjecture is false. We also provide nearly matching upper bounds for most of the problems we consider.

cs.DS↗

arXiv2016

A Dichotomy for Regular Expression Membership Testing

We study regular expression membership testing: Given a regular expression of size $m$ and a string of size $n$, decide whether the string is in the language described by the regular expression. Its classic $O(nm)$ algorithm is one of the big success stories of the 70s, which allowed pattern matching to develop into the standard tool that it is today. Many special cases of pattern matching have been studied that can be solved faster than in quadratic time. However, a systematic study of tractable cases was made possible only recently, with the first conditional lower bounds reported by Backurs and Indyk [FOCS'16]. Restricted to any "type" of homogeneous regular expressions of depth 2 or 3, they either presented a near-linear time algorithm or a quadratic conditional lower bound, with one exception known as the Word Break problem. In this paper we complete their work as follows: 1) We present two almost-linear time algorithms that generalize all known almost-linear time algorithms for special cases of regular expression membership testing. 2) We classify all types, except for the Word Break problem, into almost-linear time or quadratic time assuming the Strong Exponential Time Hypothesis. This extends the classification from depth 2 and 3 to any constant depth. 3) For the Word Break problem we give an improved $\tilde{O}(n m^{1/3} + m)$ algorithm. Surprisingly, we also prove a matching conditional lower bound for combinatorial algorithms. This establishes Word Break as the only intermediate problem. In total, we prove matching upper and lower bounds for any type of bounded-depth homogeneous regular expressions, which yields a full dichotomy for regular expression membership testing.

cs.DS↗

arXiv2015

New Unconditional Hardness Results for Dynamic and Online Problems

There has been a resurgence of interest in lower bounds whose truth rests on the conjectured hardness of well known computational problems. These conditional lower bounds have become important and popular due to the painfully slow progress on proving strong unconditional lower bounds. Nevertheless, the long term goal is to replace these conditional bounds with unconditional ones. In this paper we make progress in this direction by studying the cell probe complexity of two conjectured to be hard problems of particular importance: matrix-vector multiplication and a version of dynamic set disjointness known as Patrascu's Multiphase Problem. We give improved unconditional lower bounds for these problems as well as introducing new proof techniques of independent interest. These include a technique capable of proving strong threshold lower bounds of the following form: If we insist on having a very fast query time, then the update time has to be slow enough to compute a lookup table with the answer to every possible query. This is the first time a lower bound of this type has been proven.

cs.DS↗

arXiv2014

Towards Tight Lower Bounds for Range Reporting on the RAM

In the orthogonal range reporting problem, we are to preprocess a set of $n$ points with integer coordinates on a $U \times U$ grid. The goal is to support reporting all $k$ points inside an axis-aligned query rectangle. This is one of the most fundamental data structure problems in databases and computational geometry. Despite the importance of the problem its complexity remains unresolved in the word-RAM. On the upper bound side, three best tradeoffs exists: (1.) Query time $O(\lg \lg n + k)$ with $O(nlg^{\varepsilon}n)$ words of space for any constant $\varepsilon>0$. (2.) Query time $O((1 + k) \lg \lg n)$ with $O(n \lg \lg n)$ words of space. (3.) Query time $O((1+k)\lg^{\varepsilon} n)$ with optimal $O(n)$ words of space. However, the only known query time lower bound is $Ω(\log \log n +k)$, even for linear space data structures. All three current best upper bound tradeoffs are derived by reducing range reporting to a ball-inheritance problem. Ball-inheritance is a problem that essentially encapsulates all previous attempts at solving range reporting in the word-RAM. In this paper we make progress towards closing the gap between the upper and lower bounds for range reporting by proving cell probe lower bounds for ball-inheritance. Our lower bounds are tight for a large range of parameters, excluding any further progress for range reporting using the ball-inheritance reduction.

cs.DS↗

arXiv2014

Approximate Range Emptiness in Constant Time and Optimal Space

This paper studies the \emph{$\varepsilon$-approximate range emptiness} problem, where the task is to represent a set $S$ of $n$ points from $\{0,\ldots,U-1\}$ and answer emptiness queries of the form "$[a ; b]\cap S \neq \emptyset$ ?" with a probability of \emph{false positives} allowed. This generalizes the functionality of \emph{Bloom filters} from single point queries to any interval length $L$. Setting the false positive rate to $\varepsilon/L$ and performing $L$ queries, Bloom filters yield a solution to this problem with space $O(n \lg(L/\varepsilon))$ bits, false positive probability bounded by $\varepsilon$ for intervals of length up to $L$, using query time $O(L \lg(L/\varepsilon))$. Our first contribution is to show that the space/error trade-off cannot be improved asymptotically: Any data structure for answering approximate range emptiness queries on intervals of length up to $L$ with false positive probability $\varepsilon$, must use space $Ω(n \lg(L/\varepsilon)) - O(n)$ bits. On the positive side we show that the query time can be improved greatly, to constant time, while matching our space lower bound up to a lower order additive term. This result is achieved through a succinct data structure for (non-approximate 1d) range emptiness/reporting queries, which may be of independent interest.

cs.DS↗