Private information retrieval protocols guarantee that a user can privately and losslessly retrieve a single file from a database stored across multiple servers. In this work, we propose to simultaneously relax the conditions of perfect retrievability and privacy in order to obtain improved download rates when all files are stored uncoded on a single server. Information leakage is measured in terms of the average success probability for the server of correctly guessing the identity of the desired file. The main findings are: i) The derivation of the optimal tradeoff between download rate, distortion, and information leakage when the file size is infinite. Closed-form expressions of the optimal tradeoff for the special cases of "no-leakage" and "no-privacy" are also given. ii) A novel approach based on linear programming (LP) to construct schemes for a finite file size and an arbitrary number of files. The proposed LP approach can be leveraged to find provably optimal schemes with corresponding closed-form expressions for the rate-distortion-leakage tradeoff when the database contains at most four bits. Finally, for a database that contains 320 bits, we compare two construction methods based on the LP approach with a nonconstructive scheme downloading subsets of files using a finite-length lossy compressor based on random coding.
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The phase retrieval problem is concerned with recovering an unknown signal $\bf{x} \in \mathbb{R}^n$ from a set of magnitude-only measurements $y_j=|\langle \bf{a}_j,\bf{x} \rangle|, \; j=1,\ldots,m$. A natural least squares formulation can be used to solve this problem efficiently even with random initialization, despite its non-convexity of the loss function. One way to explain this surprising phenomenon is the benign geometric landscape: (1) all local minimizers are global; and (2) the objective function has a negative curvature around each saddle point and local maximizer. In this paper, we show that $m=O(n \log n)$ Gaussian random measurements are sufficient to guarantee the loss function of a commonly used estimator has such benign geometric landscape with high probability. This is a step toward answering the open problem given by Sun-Qu-Wright, in which the authors suggest that $O(n \log n)$ or even $O(n)$ is enough to guarantee the favorable geometric property.
The recovery of a signal from the intensity measurements with some entries being known in advance is termed as {\em affine phase retrieval}. In this paper, we prove that a natural least squares formulation for the affine phase retrieval is strongly convex on the entire space under some mild conditions, provided the measurements are complex Gaussian random vecotrs and the measurement number $m \gtrsim d \log d$ where $d$ is the dimension of signals. Based on the result, we prove that the simple gradient descent method for the affine phase retrieval converges linearly to the target solution with high probability from an arbitrary initial point. These results show an essential difference between the affine phase retrieval and the classical phase retrieval, where the least squares formulations for the classical phase retrieval are non-convex.
Non-probability sampling is prevailing in survey sampling, but ignoring its selection bias leads to erroneous inferences. We offer a unified nonparametric calibration method to estimate the sampling weights for a non-probability sample by calibrating functions of auxiliary variables in a reproducing kernel Hilbert space. The consistency and the limiting distribution of the proposed estimator are established, and the corresponding variance estimator is also investigated. Compared with existing works, the proposed method is more robust since no parametric assumption is made for the selection mechanism of the non-probability sample. Numerical results demonstrate that the proposed method outperforms its competitors, especially when the model is misspecified. The proposed method is applied to analyze the average total cholesterol of Korean citizens based on a non-probability sample from the National Health Insurance Sharing Service and a reference probability sample from the Korea National Health and Nutrition Examination Survey.
The majority of internet traffic is video content. This drives the demand for video compression in order to deliver high quality video at low target bitrates. This paper investigates the impact of adjusting the rate distortion equation on compression performance. An constant of proportionality, k, is used to modify the Lagrange multiplier used in H.265 (HEVC). Direct optimisation methods are deployed to maximise BD-Rate improvement for a particular clip. This leads to up to 21% BD-Rate improvement for an individual clip. Furthermore we use a more realistic corpus of material provided by YouTube. The results show that direct optimisation using BD-rate as the objective function can lead to further gains in bitrate savings that are not available with previous approaches.
In the past ten years there have been significant developments in optimization of transcoding parameters on a per-clip rather than per-genre basis. In our recent work we have presented per-clip optimization for the Lagrangian multiplier in Rate controlled compression, which yielded BD-Rate improvements of approximately 2\% across a corpus of videos using HEVC. However, in a video streaming application, the focus is on optimizing the rate/distortion tradeoff at a particular bitrate and not on average across a range of performance. We observed in previous work that a particular multiplier might give BD rate improvements over a certain range of bitrates, but not the entire range. Using different parameters across the range would improve gains overall. Therefore here we present a framework for choosing the best Lagrangian multiplier on a per-operating point basis across a range of bitrates. In effect, we are trying to find the para-optimal gain across bitrate and distortion for a single clip. In the experiments presented we employ direct optimization techniques to estimate this Lagrangian parameter path approximately 2,000 video clips. The clips are primarily from the YouTube-UGC dataset. We optimize both for bitrate savings as well as distortion metrics (PSNR, SSIM).
In a sports competition, a team might lose a powerful incentive to exert full effort if its final rank does not depend on the outcome of the matches still to be played. Therefore, the organiser should reduce the probability of such a situation to the extent possible. Our paper provides a classification scheme to identify these weakly (where one team is indifferent) or strongly (where both teams are indifferent) stakeless games. A statistical model is estimated to simulate the UEFA Champions League groups and compare the candidate schedules used in the 2021/22 season according to the competitiveness of the matches played in the last round(s). The option followed in four of the eight groups is found to be optimal under a wide set of parameters. Minimising the number of strongly stakeless matches is verified to be a likely goal in the computer draw of the fixture that remains hidden from the public.
Many existing algorithms for streaming geometric data analysis have been plagued by exponential dependencies in the space complexity, which are undesirable for processing high-dimensional data sets. In particular, once $d\geq\log n$, there are no known non-trivial streaming algorithms for problems such as maintaining convex hulls and L\"owner-John ellipsoids of $n$ points, despite a long line of work in streaming computational geometry since [AHV04]. We simultaneously improve these results to $\mathrm{poly}(d,\log n)$ bits of space by trading off with a $\mathrm{poly}(d,\log n)$ factor distortion. We achieve these results in a unified manner, by designing the first streaming algorithm for maintaining a coreset for $\ell_\infty$ subspace embeddings with $\mathrm{poly}(d,\log n)$ space and $\mathrm{poly}(d,\log n)$ distortion. Our algorithm also gives similar guarantees in the \emph{online coreset} model. Along the way, we sharpen results for online numerical linear algebra by replacing a log condition number dependence with a $\log n$ dependence, answering a question of [BDM+20]. Our techniques provide a novel connection between leverage scores, a fundamental object in numerical linear algebra, and computational geometry. For $\ell_p$ subspace embeddings, we give nearly optimal trade-offs between space and distortion for one-pass streaming algorithms. For instance, we give a deterministic coreset using $O(d^2\log n)$ space and $O((d\log n)^{1/2-1/p})$ distortion for $p>2$, whereas previous deterministic algorithms incurred a $\mathrm{poly}(n)$ factor in the space or the distortion [CDW18]. Our techniques have implications in the offline setting, where we give optimal trade-offs between the space complexity and distortion of subspace sketch data structures. To do this, we give an elementary proof of a "change of density" theorem of [LT80] and make it algorithmic.
The classical coding theorem in Kolmogorov complexity states that if an $n$-bit string $x$ is sampled with probability $\delta$ by an algorithm with prefix-free domain then K$(x) \leq \log(1/\delta) + O(1)$. In a recent work, Lu and Oliveira [LO21] established an unconditional time-bounded version of this result, by showing that if $x$ can be efficiently sampled with probability $\delta$ then rKt$(x) = O(\log(1/\delta)) + O(\log n)$, where rKt denotes the randomized analogue of Levin's Kt complexity. Unfortunately, this result is often insufficient when transferring applications of the classical coding theorem to the time-bounded setting, as it achieves a $O(\log(1/\delta))$ bound instead of the information-theoretic optimal $\log(1/\delta)$. We show a coding theorem for rKt with a factor of $2$. As in previous work, our coding theorem is efficient in the sense that it provides a polynomial-time probabilistic algorithm that, when given $x$, the code of the sampler, and $\delta$, it outputs, with probability $\ge 0.99$, a probabilistic representation of $x$ that certifies this rKt complexity bound. Assuming the security of cryptographic pseudorandom generators, we show that no efficient coding theorem can achieve a bound of the form rKt$(x) \leq (2 - o(1)) \cdot \log(1/\delta) +$ poly$(\log n)$. Under a weaker assumption, we exhibit a gap between efficient coding theorems and existential coding theorems with near-optimal parameters. We consider pK$^t$ complexity [GKLO22], a variant of rKt where the randomness is public and the time bound is fixed. We observe the existence of an optimal coding theorem for pK$^t$, and employ this result to establish an unconditional version of a theorem of Antunes and Fortnow [AF09] which characterizes the worst-case running times of languages that are in average polynomial-time over all P-samplable distributions.
Low-rank matrix estimation under heavy-tailed noise is challenging, both computationally and statistically. Convex approaches have been proven statistically optimal but suffer from high computational costs, especially since robust loss functions are usually non-smooth. More recently, computationally fast non-convex approaches via sub-gradient descent are proposed, which, unfortunately, fail to deliver a statistically consistent estimator even under sub-Gaussian noise. In this paper, we introduce a novel Riemannian sub-gradient (RsGrad) algorithm which is not only computationally efficient with linear convergence but also is statistically optimal, be the noise Gaussian or heavy-tailed. Convergence theory is established for a general framework and specific applications to absolute loss, Huber loss, and quantile loss are investigated. Compared with existing non-convex methods, ours reveals a surprising phenomenon of dual-phase convergence. In phase one, RsGrad behaves as in a typical non-smooth optimization that requires gradually decaying stepsizes. However, phase one only delivers a statistically sub-optimal estimator which is already observed in the existing literature. Interestingly, during phase two, RsGrad converges linearly as if minimizing a smooth and strongly convex objective function and thus a constant stepsize suffices. Underlying the phase-two convergence is the smoothing effect of random noise to the non-smooth robust losses in an area close but not too close to the truth. Lastly, RsGrad is applicable for low-rank tensor estimation under heavy-tailed noise where a statistically optimal rate is attainable with the same phenomenon of dual-phase convergence, and a novel shrinkage-based second-order moment method is guaranteed to deliver a warm initialization. Numerical simulations confirm our theoretical discovery and showcase the superiority of RsGrad over prior methods.
Common image-text joint understanding techniques presume that images and the associated text can universally be characterized by a single implicit model. However, co-occurring images and text can be related in qualitatively different ways, and explicitly modeling it could improve the performance of current joint understanding models. In this paper, we train a Cross-Modal Coherence Modelfor text-to-image retrieval task. Our analysis shows that models trained with image--text coherence relations can retrieve images originally paired with target text more often than coherence-agnostic models. We also show via human evaluation that images retrieved by the proposed coherence-aware model are preferred over a coherence-agnostic baseline by a huge margin. Our findings provide insights into the ways that different modalities communicate and the role of coherence relations in capturing commonsense inferences in text and imagery.
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