arXiv subjects
Publications and source records attributed to Gabriel Dahia.
We determine, up to a factor of $2^{o(k)}$, the number of $k$-sets $A \subset \{1, \ldots, n\}$ such that $|A + A| \leq m$, where $k = Θ(\log n)$ and $m \leq k^{1 + α}$, for small $α> 0$, answering a question of Green and Morris.
We prove, for all fixed $0 < δ< 1$, and all sufficiently large $n$, that there exists $S \subset [n]$ with $|S| \ge δn$ such that $A + B \not \subset S$ for all ${A, B \subset \mathbb{N}}$ satisfying $$\min\big\{|A|, |B|\big\} \ge \big(3 + o(1)\big) \frac{\log n }{ \log (1 / δ)}.$$ A very recent result of Hernández and Hetzel shows that our bound is sharp up to a factor of 3, and together our results settle a conjecture of Kra, Moreira, Richter, and Robertson. In fact, we prove that a $δ$-dense random subset of $[n]$ is a valid choice for $S$ with high probability, and that one can take $n^{-α} \le δ\le 1 - c$ where $c > 0$ is fixed and $α> 0$ depends only on the $o(1)$ error, answering another question of the same authors in a strong form.
The induced Ramsey number $R_{\mathrm{ind}}(H; r)$ of a graph $H$ is the minimum number $N$ such that there exists a graph with $N$ vertices for which all $r$-colourings of its edges contain a monochromatic induced copy of $H$. Our main result is the existence of a constant $C > 0$ such that, for every graph $H$ on $k$ vertices, these numbers satisfy \begin{equation*} R_{\mathrm{ind}}(H; r) \le r^{C r k}. \end{equation*} When $r = 2$, this resolves a conjecture of Erdős from 1975. For $r > 2$, it answers a question of Conlon, Fox and Sudakov in a strong form.
The Cayley sum graph $Γ_A$ of a set $A \subseteq \mathbb{Z}_n$ is defined to have vertex set $\mathbb{Z}_n$ and an edge between two distinct vertices $x, y \in \mathbb{Z}_n$ if $x + y \in A$. Green and Morris proved that if the set $A$ is a $p$-random subset of $\mathbb{Z}_n$ with $p = 1/2$, then the independence number of $Γ_A$ is asymptotically equal to $α(G(n, 1/2))$ with high probability. Our main theorem is the first extension of their result to $p = o(1)$: we show that, with high probability, $$α(Γ_A) = (1 + o(1)) α(G(n, p))$$ as long as $p \ge (\log n)^{-1/80}$. One of the tools in our proof is a geometric-flavoured theorem that generalises Freĭman's lemma, the classical lower bound on the size of high dimensional sumsets. We also give a short proof of this result up to a constant factor; this version yields a much simpler proof of our main theorem at the expense of a worse constant.
The $n$-dimensional random twisted hypercube $\mathbf{G}_n$ is constructed recursively by taking two instances of $\mathbf{G}_{n-1}$, with any joint distribution, and adding a random perfect matching between their vertex sets. Benjamini, Dikstein, Gross, and Zhukovskii showed that its diameter is $O(n\log \log \log n/\log \log n)$ with high probability and at least ${(n - 1)/ \log_2 n}$. We improve their upper bound by showing that $$\operatorname{diam}(\mathbf{G}_n) = \big(1 + o(1)\big) \frac{n}{\log_2 n}$$ with high probability.
We propose a method that can perform one-class classification given only a small number of examples from the target class and none from the others. We formulate the learning of meaningful features for one-class classification as a meta-learning problem in which the meta-training stage repeatedly simulates one-class classification, using the classification loss of the chosen algorithm to learn a feature representation. To learn these representations, we require only multiclass data from similar tasks. We show how the Support Vector Data Description method can be used with our method, and also propose a simpler variant based on Prototypical Networks that obtains comparable performance, indicating that learning feature representations directly from data may be more important than which one-class algorithm we choose. We validate our approach by adapting few-shot classification datasets to the few-shot one-class classification scenario, obtaining similar results to the state-of-the-art of traditional one-class classification, and that improves upon that of one-class classification baselines employed in the few-shot setting. Our code is available at https://github.com/gdahia/meta_occ
High-resolution fingerprint recognition often relies on sophisticated matching algorithms based on hand-crafted keypoint descriptors, with pores being the most common keypoint choice. Our method is the opposite of the prevalent approach: we use instead a simple matching algorithm based on robust local pore descriptors that are learned from the data using a CNN. In order to train this CNN in a fully supervised manner, we describe how the automatic alignment of fingerprint images can be used to obtain the required training annotations, which are otherwise missing in all publicly available datasets. This improves the state-of-the-art recognition results for both partial and full fingerprints in a public benchmark. To confirm that the observed improvement is due to the adoption of learned descriptors, we conduct an ablation study using the most successful pore descriptors previously used in the literature. All our code is available at https://github.com/gdahia/high-res-fingerprint-recognition
In this work, we investigate if previously proposed CNNs for fingerprint pore detection overestimate the number of required model parameters for this task. We show that this is indeed the case by proposing a fully convolutional neural network that has significantly fewer parameters. We evaluate this model using a rigorous and reproducible protocol, which was, prior to our work, not available to the community. Using our protocol, we show that the proposed model, when combined with post-processing, performs better than previous methods, albeit being much more efficient. All our code is available at https://github.com/gdahia/fingerprint-pore-detection