We derive the first explicit bounds for the spectral gap of a random walk Metropolis algorithm on $R^d$ for any value of the proposal variance, which when scaled appropriately recovers the correct $d^{-1}$ dependence on dimension for suitably regular invariant distributions. We also obtain explicit bounds on the ${\rm L}^2$-mixing time for a broad class of models. In obtaining these results, we refine the use of isoperimetric profile inequalities to obtain conductance profile bounds, which also enable the derivation of explicit bounds in a much broader class of models. We also obtain similar results for the preconditioned Crank--Nicolson Markov chain, obtaining dimension-independent bounds under suitable assumptions.
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$\textbf{Background and aims}$: Artificial Intelligence (AI) Computer-Aided Detection (CADe) is commonly used for polyp detection, but data seen in clinical settings can differ from model training. Few studies evaluate how well CADe detectors perform on colonoscopies from countries not seen during training, and none are able to evaluate performance without collecting expensive and time-intensive labels. $\textbf{Methods}$: We trained a CADe polyp detector on Israeli colonoscopy videos (5004 videos, 1106 hours) and evaluated on Japanese videos (354 videos, 128 hours) by measuring the True Positive Rate (TPR) versus false alarms per minute (FAPM). We introduce a colonoscopy dissimilarity measure called "MAsked mediCal Embedding Distance" (MACE) to quantify differences between colonoscopies, without labels. We evaluated CADe on all Japan videos and on those with the highest MACE. $\textbf{Results}$: MACE correctly quantifies that narrow-band imaging (NBI) and chromoendoscopy (CE) frames are less similar to Israel data than Japan whitelight (bootstrapped z-test, |z| > 690, p < $10^{-8}$ for both). Despite differences in the data, CADe performance on Japan colonoscopies was non-inferior to Israel ones without additional training (TPR at 0.5 FAPM: 0.957 and 0.972 for Israel and Japan; TPR at 1.0 FAPM: 0.972 and 0.989 for Israel and Japan; superiority test t > 45.2, p < $10^{-8}$). Despite not being trained on NBI or CE, TPR on those subsets were non-inferior to Japan overall (non-inferiority test t > 47.3, p < $10^{-8}$, $\delta$ = 1.5% for both). $\textbf{Conclusion}$: Differences that prevent CADe detectors from performing well in non-medical settings do not degrade the performance of our AI CADe polyp detector when applied to data from a new country. MACE can help medical AI models internationalize by identifying the most "dissimilar" data on which to evaluate models.
For a graph $ G = (V, E) $ with a vertex set $ V $ and an edge set $ E $, a function $ f : V \rightarrow \{0, 1, 2, . . . , diam(G)\} $ is called a \emph{broadcast} on $ G $. For each vertex $ u \in V $, if there exists a vertex $ v $ in $ G $ (possibly, $ u = v $) such that $ f (v) > 0 $ and $ d(u, v) \leq f (v) $, then $ f $ is called a dominating broadcast on $ G $. The cost of the dominating broadcast $f$ is the quantity $ \sum_{v\in V}f(v) $. The minimum cost of a dominating broadcast is the broadcast domination number of $G$, denoted by $ \gamma_{b}(G) $. A multipacking is a set $ S \subseteq V $ in a graph $ G = (V, E) $ such that for every vertex $ v \in V $ and for every integer $ r \geq 1 $, the ball of radius $ r $ around $ v $ contains at most $ r $ vertices of $ S $, that is, there are at most $ r $ vertices in $ S $ at a distance at most $ r $ from $ v $ in $ G $. The multipacking number of $ G $ is the maximum cardinality of a multipacking of $ G $ and is denoted by $ mp(G) $. We show that, for any connected chordal graph $G$, $\gamma_{b}(G)\leq \big\lceil{\frac{3}{2} mp(G)\big\rceil}$. We also show that $\gamma_b(G)-mp(G)$ can be arbitrarily large for connected chordal graphs by constructing an infinite family of connected chordal graphs such that the ratio $\gamma_b(G)/mp(G)=10/9$, with $mp(G)$ arbitrarily large. Moreover, we show that $\gamma_{b}(G)\leq \big\lfloor{\frac{3}{2} mp(G)+2\delta\big\rfloor} $ holds for all $\delta$-hyperbolic graphs. In addition, we provide a polynomial-time algorithm to construct a multipacking of a $\delta$-hyperbolic graph $G$ of size at least $ \big\lceil{\frac{2mp(G)-4\delta}{3} \big\rceil} $.
Advances in survival analysis have facilitated unprecedented flexibility in data modeling, yet there remains a lack of tools for graphically illustrating the influence of continuous covariates on predicted survival outcomes. We propose the utilization of a colored contour plot to depict the predicted survival probabilities over time, and provide a Shiny app and R package as implementations of this tool. Our approach is capable of supporting conventional models, including the Cox and Fine-Gray models. However, its capability shines when coupled with cutting-edge machine learning models such as random survival forests and deep neural networks.
Spatial data can come in a variety of different forms, but two of the most common generating models for such observations are random fields and point processes. Whilst it is known that spectral analysis can unify these two different data forms, specific methodology for the related estimation is yet to be developed. In this paper, we solve this problem by extending multitaper estimation, to estimate the spectral density matrix function for multivariate spatial data, where processes can be any combination of either point processes or random fields. We discuss finite sample and asymptotic theory for the proposed estimators, as well as specific details on the implementation, including how to perform estimation on non-rectangular domains and the correct implementation of multitapering for processes sampled in different ways, e.g. continuously vs on a regular grid.
A multifold $1$-perfect code ($1$-perfect code for list decoding) in any graph is a set $C$ of vertices such that every vertex of the graph is at distance not more than $1$ from exactly $\mu$ elements of $C$. In $q$-ary Hamming graphs, where $q$ is a prime power, we characterize all parameters of multifold $1$-perfect codes and all parameters of additive multifold $1$-perfect codes. In particular, we show that additive multifold $1$-perfect codes are related to special multiset generalizations of spreads, multispreads, and that multispreads of parameters corresponding to multifold $1$-perfect codes always exist. Keywords: perfect codes, multifold packing, multiple covering, list-decoding codes, additive codes, spreads, multispreads, completely regular codes, intriguing sets.
In a recently developed variational discretization scheme for second order initial value problems ( J. Comput. Phys. 498, 112652 (2024) ), it was shown that the Noether charge associated with time translation symmetry is exactly preserved in the interior of the simulated domain. The obtained solution also fulfils the naively discretized equations of motions inside the domain, except for the last two grid points. Here we provide an explanation for the deviations at the boundary as stemming from the Lagrange multipliers used to implement initial and connection conditions. We show explicitly that the Noether charge including the boundary corrections is exactly preserved at its continuum value over the whole simulation domain, including the boundary points.
We present a comprehensive analysis of the implications of artificial latency in the Proposer-Builder Separation framework on the Ethereum network. Focusing on the MEV-Boost auction system, we analyze how strategic latency manipulation affects Maximum Extractable Value yields and network integrity. Our findings reveal both increased profitability for node operators and significant systemic challenges, including heightened network inefficiencies and centralization risks. We empirically validates these insights with a pilot that Chorus One has been operating on Ethereum mainnet. We demonstrate the nuanced effects of latency on bid selection and validator dynamics. Ultimately, this research underscores the need for balanced strategies that optimize Maximum Extractable Value capture while preserving the Ethereum network's decentralization ethos.
The existence of $q$-ary linear complementary pairs (LCPs) of codes with $q> 2$ has been completely characterized so far. This paper gives a characterization for the existence of binary LCPs of codes. As a result, we solve an open problem proposed by Carlet $et~al.$ (IEEE Trans. Inf. Theory 65(3): 1694-1704, 2019) and a conjecture proposed by Choi $et~al.$ (Cryptogr. Commun. 15(2): 469-486, 2023).
A class of (block) rational Krylov subspace based projection method for solving large-scale continuous-time algebraic Riccati equation (CARE) $0 = \mathcal{R}(X) := A^HX + XA + C^HC - XBB^HX$ with a large, sparse $A$ and $B$ and $C$ of full low rank is proposed. The CARE is projected onto a block rational Krylov subspace $\mathcal{K}_j$ spanned by blocks of the form $(A^H+ s_kI)C^H$ for some shifts $s_k, k = 1, \ldots, j.$ The considered projections do not need to be orthogonal and are built from the matrices appearing in the block rational Arnoldi decomposition associated to $\mathcal{K}_j.$ The resulting projected Riccati equation is solved for the small square Hermitian $Y_j.$ Then the Hermitian low-rank approximation $X_j = Z_jY_jZ_j^H$ to $X$ is set up where the columns of $Z_j$ span $\mathcal{K}_j.$ The residual norm $\|R(X_j )\|_F$ can be computed efficiently via the norm of a readily available $2p \times 2p$ matrix. We suggest to reduce the rank of the approximate solution $X_j$ even further by truncating small eigenvalues from $X_j.$ This truncated approximate solution can be interpreted as the solution of the Riccati residual projected to a subspace of $\mathcal{K}_j.$ This gives us a way to efficiently evaluate the norm of the resulting residual. Numerical examples are presented.
By approximating posterior distributions with weighted samples, particle filters (PFs) provide an efficient mechanism for solving non-linear sequential state estimation problems. While the effectiveness of particle filters has been recognised in various applications, their performance relies on the knowledge of dynamic models and measurement models, as well as the construction of effective proposal distributions. An emerging trend involves constructing components of particle filters using neural networks and optimising them by gradient descent, and such data-adaptive particle filtering approaches are often called differentiable particle filters. Due to the expressiveness of neural networks, differentiable particle filters are a promising computational tool for performing inference on sequential data in complex, high-dimensional tasks, such as vision-based robot localisation. In this paper, we review recent advances in differentiable particle filters and their applications. We place special emphasis on different design choices for key components of differentiable particle filters, including dynamic models, measurement models, proposal distributions, optimisation objectives, and differentiable resampling techniques.
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