اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدStructure-adaptive manifold estimation
138
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We consider a problem of manifold estimation from noisy observations. Many manifold learning procedures locally approximate a manifold by a weighted average over a small neighborhood. However, in the presence of large noise, the assigned weights become so corrupted that the averaged estimate shows very poor performance. We suggest a novel computationally efficient structure-adaptive procedure which simultaneously reconstructs a smooth manifold and estimates projections of the point cloud onto this manifold. The proposed approach iteratively refines the weights on each step, using the structural information obtained at previous steps. After several iterations, we obtain nearly oracle weights, so that the final estimates are nearly efficient even in the presence of relatively large noise. In our theoretical study we establish tight lower and upper bounds proving asymptotic optimality of the method for manifold estimation under the Hausdorff loss, provided that the noise degrades to zero fast enough.
قيم البحث
اقرأ أيضاً
Clustering methods seek to partition data such that elements are more similar to elements in the same cluster than to elements in different clusters. The main challenge in this task is the lack of a unified definition of a cluster, especially for hig
h dimensional data. Different methods and approaches have been proposed to address this problem. This paper continues the study originated by Efimov, Adamyan and Spokoiny (2019) where a novel approach to adaptive nonparametric clustering called Adaptive Weights Clustering (AWC) was offered. The method allows analyzing high-dimensional data with an unknown number of unbalanced clusters of arbitrary shape under very weak modeling assumptions. The procedure demonstrates a state-of-the-art performance and is very efficient even for large data dimension D. However, the theoretical study in Efimov, Adamyan and Spokoiny (2019) is very limited and did not really address the question of efficiency. This paper makes a significant step in understanding the remarkable performance of the AWC procedure, particularly in high dimension. The approach is based on combining the ideas of adaptive clustering and manifold learning. The manifold hypothesis means that high dimensional data can be well approximated by a d-dimensional manifold for small d helping to overcome the curse of dimensionality problem and to get sharp bounds on the cluster separation which only depend on the intrinsic dimension d. We also address the problem of parameter tuning. Our general theoretical results are illustrated by some numerical experiments.
We study distributed estimation methods under communication constraints in a distributed version of the nonparametric random design regression model. We derive minimax lower bounds and exhibit methods that attain those bounds. Moreover, we show that adaptive estimation is possible in this setting.
This paper is concerned with density estimation of directional data on the sphere. We introduce a procedure based on thresholding on a new type of spherical wavelets called {it needlets}. We establish a minimax result and prove its optimality. We are
motivated by astrophysical applications, in particular in connection with the analysis of ultra high energy cosmic rays.
We consider the problem of undirected graphical model inference. In many applications, instead of perfectly recovering the unknown graph structure, a more realistic goal is to infer some graph invariants (e.g., the maximum degree, the number of conne
cted subgraphs, the number of isolated nodes). In this paper, we propose a new inferential framework for testing nested multiple hypotheses and constructing confidence intervals of the unknown graph invariants under undirected graphical models. Compared to perfect graph recovery, our methods require significantly weaker conditions. This paper makes two major contributions: (i) Methodologically, for testing nested multiple hypotheses, we propose a skip-down algorithm on the whole family of monotone graph invariants (The invariants which are non-decreasing under addition of edges). We further show that the same skip-down algorithm also provides valid confidence intervals for the targeted graph invariants. (ii) Theoretically, we prove that the length of the obtained confidence intervals are optimal and adaptive to the unknown signal strength. We also prove generic lower bounds for the confidence interval length for various invariants. Numerical results on both synthetic simulations and a brain imaging dataset are provided to illustrate the usefulness of the proposed method.
We investigate whether in a distributed setting, adaptive estimation of a smooth function at the optimal rate is possible under minimal communication. It turns out that the answer depends on the risk considered and on the number of servers over which
the procedure is distributed. We show that for the $L_infty$-risk, adaptively obtaining optimal rates under minimal communication is not possible. For the $L_2$-risk, it is possible over a range of regularities that depends on the relation between the number of local servers and the total sample size.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
