Marginal MAP inference involves making MAP predictions in systems defined with latent variables or missing information. It is significantly more difficult than pure marginalization and MAP tasks, for which a large class of efficient and convergent variational algorithms, such as dual decomposition, exist. In this work, we generalize dual decomposition to a generic power sum inference task, which includes marginal MAP, along with pure marginalization and MAP, as special cases. Our method is based on a block coordinate descent algorithm on a new convex decomposition bound, that is guaranteed to converge monotonically, and can be parallelized efficiently. We demonstrate our approach on marginal MAP queries defined on real-world problems from the UAI approximate inference challenge, showing that our framework is faster and more reliable than previous methods.
While our focus is on variational bounds @cite_5 @cite_15 , there are many non-variational approaches for marginal MAP as well. provide upper bounds on marginal MAP by reordering the order in which variables are eliminated, and using exact inference in the reordered join-tree; however, this is exponential in the size of the (unconstrained) treewidth, and can easily become intractable. give an approximation closely related to mini-bucket to bound the marginal MAP; however, unlike (weighted) mini-bucket, these bounds cannot be improved iteratively. The same is true for the algorithm of , which also has a strong dependence on treewidth. Other examples of marginal MAP algorithms include local search [e.g.,][] park2004complexity and Markov chain Monte Carlo methods [e.g.,][] doucet02,yuan04 .
@cite_5 @cite_15
2407138673 1959656505 2127856808 1968429800 2571471359 2151239055 2158842374 2963702702 2069278600 1952648253 2508647357 2098322943 1993855803 1599871777 1539230104 2108619558 2745919252 2161431029 1976930960 2124335859
Marginal MAP problems are known to be very difficult tasks for graphical models and are so far solved exactly by systematic search guided by a join-tree upper bound. In this paper, we develop new AND OR branch and bound algorithms for marginal MAP that use heuristics extracted from weighted mini-buckets enhanced with message-passing updates. We demonstrate the effectiveness of the resulting search algorithms against previous join-tree based approaches, which we also extend to accommodate high induced width models, through extensive empirical evaluations. Our results show not only orders-of-magnitude improvements over the state-of-the-art, but also the ability to solve problem instances well beyond the reach of previous approaches. The MAP (maximum a posteriori hypothesis) problem in Bayesian networks is to find the most likely states of a set of variabls given partial evidence on the complement of that set. Standard structure-based inference methods for finding exact solutions to MAP, such as variable elimination and join-tree algorithms, have complexities that are exponential in the constrained treewidth of the network. A more recent algorithm, proposed by Park and Darwiche, is exponential only in the treewidth and has been shown to handle networks whose constrained treewidth is quite high. In this paper we present a new algorithm for exact MAP that is not necessarily limited in scalability even by the treewidth. This is achieved by leveraging recent advances in compilation of Bayesian networks into arithmetic circuits, which can circumvent treewidth-imposed limits by exploiting the local structure present in the network. Specifically, we implement a branch-and-bound search where the bounds are computed using linear-time operations on the compiled arithmetic circuit. On networks with local structure, we observe orders-of-magnitude improvements over the algorithm of Park and Darwiche. In particular, we are able to efficiently solve many problems where the latter algorithm runs out of memory because of high treewidth. Until recently, the lack of ground truth data has hindered the application of discriminative structured prediction techniques to the stereo problem. In this paper we use ground truth data sets that we have recently constructed to explore different model structures and parameter learning techniques. To estimate parameters in Markov random fields (MRFs) via maximum likelihood one usually needs to perform approximate probabilistic inference. Conditional random fields (CRFs) are discriminative versions of traditional MRFs. We explore a number of novel CRF model structures including a CRF for stereo matching with an explicit occlusion model. CRFs require expensive inference steps for each iteration of optimization and inference is particularly slow when there are many discrete states. We explore belief propagation, variational message passing and graph cuts as inference methods during learning and compare with learning via pseudolikelihood. To accelerate approximate inference we have developed a new method called sparse variational message passing which can reduce inference time by an order of magnitude with negligible loss in quality. Learning using sparse variational message passing improves upon previous approaches using graph cuts and allows efficient learning over large data sets when energy functions violate the constraints imposed by graph cuts. We study the quantitative geometry of graphs in terms of their genus, using the structure of certain "cut graphs," i.e. subgraphs whose removal leaves a planar graph. In particular, we give optimal bounds for random partitioning schemes, as well as various types of embeddings. Using these geometric primitives, we present exponentially improved dependence on genus for a number of problems like approximate max-flow min-cut theorems, approximations for uniform and nonuniform Sparsest Cut, treewidth approximation, Laplacian eigenvalue bounds, and Lipschitz extension theorems and related metric labeling problems. We list here a sample of these improvements. All the following statements refer to graphs of genus g, unless otherwise noted. • We show that such graphs admit an O(log g)-approximate multi-commodity max-flow min-cut theorem for the case of uniform demands. This bound is optimal, and improves over the previous bound of O(g) [KPR93, FT03]. For general demands, we show that the worst possible gap is O(log g + CP), where CP is the gap for planar graphs. This dependence is optimal, and already yields a bound of O(log g + √log n), improving over the previous bound of O(√g log n) [KLMN04]. • We give an O(√log g)-approximation for the uniform Sparsest Cut, balanced vertex separator, and treewidth problems, improving over the previous bound of O(g) [FHL05]. • If a graph G has genus g and maximum degree D, we show that the kth Laplacian eigenvalue of G is (log g)2 · O(kgD n), improving over the previous bound of g2·O(kgD n) [KLPT09]. There is a lower bound of Ω(kgD n), making this result almost tight. • We show that if (X, d) is the shortest-path metric on a graph of genus g and S ⊆ X, then every L-Lipschitz map f: S → Z into a Banach space Z admits an O(L log g)-Lipschitz extension f: X → Z. This improves over the previous bound of O(Lg) [LN05], and compares to a lower bound of Ω(L√log g). In a related way, we show that there is an O(log g)-approximation for the 0-extension problem on such graphs, improving over the previous O(g) bound. • We show that every n-vertex shortest-path metric on a graph of genus g embeds into L2 with distortion O(log g + √log n), improving over the previous bound of O(√g log n). Our result is asymptotically optimal for every dependence g = g(n). We present two algorithms for dynamically maintaining a spanning forest of a graph undergoing edge insertions and deletions. Our algorithms guarantee worst-case update time and work against an adaptive adversary, meaning that an edge update can depend on previous outputs of the algorithms. We provide the first polynomial improvement over the long-standing O(√n) bound of [Frederickson STOC'84, Eppstein, Galil, Italiano and Nissenzweig FOCS'92] for such type of algorithms. The previously best improvement was O(√n (loglogn)2 logn) [Kejlberg-Rasmussen, Kopelowitz, Pettie and Thorup ESA'16]. We note however that these bounds were obtained by deterministic algorithms while our algorithms are randomized. Our first algorithm is Monte Carlo and guarantees an O(n0.4+o(1)) worst-case update time, where the o(1) term hides the O(√loglogn logn) factor. Our second algorithm is Las Vegas and guarantee an O(n0.49306) worst-case update time with high probability. Algorithms with better update time either needed to assume that the adversary is oblivious (e.g. [Kapron, King and Mountjoy SODA'13]) or can only guarantee an amortized update time. Our second result answers an open problem by To the best of our knowledge, our algorithms are among a few non-trivial randomized dynamic algorithms that work against adaptive adversaries. The key to our results is a decomposition of graphs into subgraphs that either have high expansion or sparse. This decomposition serves as an interface between recent developments on (static) flow computation and many old ideas in dynamic graph algorithms: On the one hand, we can combine previous dynamic graph techniques to get faster dynamic spanning forest algorithms if such decomposition is given. On the other hand, we can adapt flow-related techniques (e.g. those from [Khandekar, Rao and Vazirani STOC'06], [Peng SODA'16], and [Orecchia and Zhu SODA'14]) to maintain such decomposition. To the best of our knowledge, this is the first time these flow techniques are used in fully dynamic graph algorithms. Given an arc-weighted directed graph G = (V, A, spl lscr ) and a pair of nodes s, t, we seek to find an s-t walk of length at most B that maximizes some given function f of the set of nodes visited by the walk. The simplest case is when we seek to maximize the number of nodes visited: this is called the orienteering problem. Our main result is a quasi-polynomial time algorithm that yields an O(log OPT) approximation for this problem when f is a given submodular set function. We then extend it to the case when a node v is counted as visited only if the walk reaches v in its time window [R(v), D(v)]. We apply the algorithm to obtain several new results. First, we obtain an O(log OPT) approximation for a generalization of the orienteering problem in which the profit for visiting each node may vary arbitrarily with time. This captures the time window problem considered earlier for which, even in undirected graphs, the best approximation ratio known [Bansal, (2004)] is O(log sup 2 OPT). The second application is an O(log sup 2 k) approximation for the k-TSP problem in directed graphs (satisfying asymmetric triangle inequality). This is the first non-trivial approximation algorithm for this problem. The third application is an O(log sup 2 k) approximation (in quasi-poly time) for the group Steiner problem in undirected graphs where k is the number of groups. This improves earlier ratios (Garg, ) by a logarithmic factor and almost matches the inapproximability threshold on trees (Halperin and Krauthgamer, 2003). This connection to group Steiner trees also enables us to prove that the problem we consider is hard to approximate to a ratio better than spl Omega (log sup 1- spl epsi OPT), even in undirected graphs. Even though our algorithm runs in quasi-poly time, we believe that the implications for the approximability of several basic optimization problems are interesting. Many state-of-the-art segmentation algorithms rely on Markov or Conditional Random Field models designed to enforce spatial and global consistency constraints. This is often accomplished by introducing additional latent variables to the model, which can greatly increase its complexity. As a result, estimating the model parameters or computing the best maximum a posteriori (MAP) assignment becomes a computationally expensive task. In a series of experiments on the PASCAL and the MSRC datasets, we were unable to find evidence of a significant performance increase attributed to the introduction of such constraints. On the contrary, we found that similar levels of performance can be achieved using a much simpler design that essentially ignores these constraints. This more simple approach makes use of the same local and global features to leverage evidence from the image, but instead directly biases the preferences of individual pixels. While our investigation does not prove that spatial and consistency constraints are not useful in principle, it points to the conclusion that they should be validated in a larger context. We prove tight upper bounds on the logarithmic derivative of the independence and matching polynomials of d-regular graphs. For independent sets, this theorem is a strengthening of Kahn's result that a disjoint union of copies of Kd;d maximizes the number of independent sets of a bipartite d-regular graph, Galvin and Tet ali's result that the independence polynomial is maximized by the same, and Zhao's extension of both results to all d-regular graphs. For matchings, this shows that the matching polynomial and the total number of matchings of a d-regular graph are maximized by a union of copies of Kd;d. Using this we prove the asymptotic upper matching conjecture of Friedland, Krop, Lundow, and Markstrom. In probabilistic language, our main theorems state that for all d-regular graphs and all �, the occupancy fraction of the hard-core model and the edge occupancy fraction of the monomer-dimer model with fugacity � are maximized by Kd;d. Our method involves constrained optimization problems over distributions of random variables and applies to all d-regular graphs directly, without a reduction to the bipartite case. Using a variant of the method we prove a lower bound on the occupancy fraction of the hard-core model on any d-regular, vertex-transitive, bipartite graph: the occupancy fraction of such a graph is strictly greater than the occupancy fraction of the unique translationinvariant hard-core measure on the infinite d-regular tree We present a new efficient algorithm for the search version of the approximate Closest Vector Problem with Preprocessing (CVPP). Our algorithm achieves an approximation factor of O(n sqrt log n ), improving on the previous best of O(n^ 1.5 ) due to Lag arias, Lenstra, and Schnorr hkzbabai . We also show, somewhat surprisingly, that only O(n) vectors of preprocessing advice are sufficient to solve the problem (with the slightly worse approximation factor of O(n)). We remark that this still leaves a large gap with respect to the decisional version of CVPP, where the best known approximation factor is O(sqrt n log n ) due to Aharonov and Regev AharonovR04 . To achieve these results, we show a reduction to the same problem restricted to target points that are close to the lattice and a more efficient reduction to a harder problem, Bounded Distance Decoding with preprocessing (BDDP). Combining either reduction with the previous best-known algorithm for BDDP by Liu, Lyubashevsky, and Micciancio LiuLM06 gives our main result. In the setting of CVP without preprocessing, we also give a reduction from (1+eps)gamma approximate CVP to gamma approximate CVP where the target is at distance at most 1+1 eps times the minimum distance (the length of the shortest non-zero vector) which relies on the lattice sparsification techniques of Dadush and Kun DadushK13 . As our final and most technical contribution, we present a substantially more efficient variant of the LLM algorithm (both in terms of run-time and amount of preprocessing advice), and via an improved analysis, show that it can decode up to a distance proportional to the reciprocal of the smoothing parameter of the dual lattice MR04 . We show that this is never smaller than the LLM decoding radius, and that it can be up to an wide tilde Omega (sqrt n ) factor larger. Y. Bartal (1996, 1998) gave a randomized polynomial time algorithm that given any n point metric G, constructs a tree T such that the expected stretch (distortion) of any edge is at most O (log n log log n). His result has found several applications and in particular has resulted in approximation algorithms for many graph optimization problems. However approximation algorithms based on his result are inherently randomized. In this paper we derandomize the use of Bartal's algorithm in the design of approximation algorithms. We give an efficient polynomial time algorithm that given a finite n point metric G, constructs O(n log n) trees and a probability distribution spl mu on them such that the expected stretch of any edge of G in a tree chosen according to spl mu is at most O(log n log log n). Our result establishes that finite metrics can be probabilistically approximated by a small number of tree metrics. We obtain the first deterministic approximation algorithms for buy-at-bulk network design and vehicle routing; in addition we subsume results from our earlier work on derandomization. Our main result is obtained by a novel view of probabilistic approximation of metric spaces as a deterministic optimization problem via linear programming. Spanners, emulators, and approximate distance oracles can be viewed as lossy compression schemes that represent an unweighted graph metric in small space, say O(n1+Δ) bits. There is an inherent tradeoff between the sparsity parameter Δ and the stretch function f of the compression scheme, but the qualitative nature of this tradeoff has remained a persistent open problem. It has been known for some time that when Δ ≥ 1 3 there are schemes with constant additive stretch (distance d is stretched to at most f(d) = d + O(1)), and recent results of Abboud and Bodwin show that when Δ In this paper we show that the lower bound of Abboud and Bodwin is just the first step in a hierarchy of lower bounds that characterize the asymptotic behavior of the optimal stretch function f for sparsity parameter Δ ∈ (0,1 3). Specifically, for any integer k ≥ 2, any compression scheme with size [EQUATION] has a sublinear additive stretch function f: f(d) = d + Ω(d1−1 k). This lower bound matches Thorup and Zwick's (2006) construction of sublinear additive emulators. It also shows that Elkin and Peleg's (1 + ϵ, β)-spanners have an essentially optimal tradeoff between Δ, ϵ, and β, and that the sublinear additive spanners of Pettie (2009) and Chechik (2013) are not too far from optimal. To complement these lower bounds we present a new construction of (1 + ϵ, O(k ϵ)k−1)-spanners with size [EQUATION], where hk Our lower bound technique exhibits several interesting degrees of freedom in the framework of Abboud and Bodwin. By carefully exploiting these freedoms, we are able to obtain lower bounds for several related combinatorial objects. We get lower bounds on the size of (β, ϵ)-hopsets, matching Elkin and Neiman's construction (2016), and lower bounds on shortcutting sets for digraphs that preserve the transitive closure. Our lower bound simplifies Hesse's (2003) refutation of Thorup's conjecture (1992), which stated that adding a linear number of shortcuts suffices to reduce the diameter to polylogarithmic. Finally, we show matching upper and lower bounds for graph compression schemes that work for graph metrics with girth at least 2γ + 1. One consequence is that 's (2010) additive O(γ)-spanners with size [EQUATION] cannot be improved in the exponent. We prove that any graph excluding Kr as a minor has can be partitioned into clusters of diameter at most Δ while removing at most O(r Δ) fraction of the edges. This improves over the results of Fakcharoenphol and Talwar, who building on the work of Klein, Plotkin and Rao gave a partitioning that required to remove O(r2 Δ) fraction of the edges. Our result is obtained by a new approach that relates the topological properties (excluding a minor) of a graph to its geometric properties (the induced shortest path metric). Specifically, we show that techniques used by Andreae in his investigation of the cops and robbers game on graphs excluding a fixed minor, can be used to construct padded decompositions of the metrics induced by such graphs. In particular, we get probabilistic partitions with padding parameter O(r) and strong-diameter partitions with padding parameter O(r2) for Kr-free graphs, O(k) for treewidth-k graphs, and O(log g) for graphs with genus g. In this paper, we attempt to approximate and index a d- dimensional (d ≥ 1) spatio-temporal trajectory with a low order continuous polynomial. There are many possible ways to choose the polynomial, including (continuous)Fourier transforms, splines, non-linear regressino, etc. Some of these possiblities have indeed been studied beofre. We hypothesize that one of the best possibilities is the polynomial that minimizes the maximum deviation from the true value, which is called the minimax polynomial. Minimax approximation is particularly meaningful for indexing because in a branch-and-bound search (i.e., for finding nearest neighbours), the smaller the maximum deviation, the more pruning opportunities there exist. However, in general, among all the polynomials of the same degree, the optimal minimax polynomial is very hard to compute. However, it has been shown thta the Chebyshev approximation is almost identical to the optimal minimax polynomial, and is easy to compute [16]. Thus, in this paper, we explore how to use the Chebyshev polynomials as a basis for approximating and indexing d-dimenstional trajectories.The key analytic result of this paper is the Lower Bounding Lemma. that is, we show that the Euclidean distance between two d-dimensional trajectories is lower bounded by the weighted Euclidean distance between the two vectors of Chebyshev coefficients. this lemma is not trivial to show, and it ensures that indexing with Chebyshev cofficients aedmits no false negatives. To complement that analystic result, we conducted comprehensive experimental evaluation with real and generated 1-dimensional to 4-dimensional data sets. We compared the proposed schem with the Adaptive Piecewise Constant Approximation (APCA) scheme. Our preliminary results indicate that in all situations we tested, Chebyshev indexing dominates APCA in pruning power, I O and CPU costs. Breiman (2001a,b) has recently developed an ensemble classification and regression approach that displayed outstanding performance with regard prediction error on a suite of benchmark datasets. As the base constituents of the ensemble are tree-structured predictors, and since each of these is constructed using an injection of randomness, the method is called ‘random forests’. That the exceptional performance is attained with seemingly only a single tuning parameter, to which sensitivity is minimal, makes the methodology all the more remarkable. The individual trees comprising the forest are all grown to maximal depth. While this helps with regard bias, there is the familiar tradeoff with variance. However, these variability concerns were potentially obscured because of an interesting feature of those benchmarking datasets extracted from the UCI machine learning repository for testing: all these datasets are hard to overfit using tree-structured methods. This raises issues about the scope of the repository. With this as motivation, and coupled with experience from boosting methods, we revisit the formulation of random forests and investigate prediction performance on real-world and simulated datasets for which maximally sized trees do overfit. These explorations reveal that gains can be realized by additional tuning to regulate tree size via limiting the number of splits and or the size of nodes for which splitting is allowed. Nonetheless, even in these settings, good performance for random forests can be attained by using larger (than default) primary tuning parameter values. We propose an algorithm to improve the quality of depth-maps used for Multi-View Stereo (MVS). Many existing MVS techniques make use of a two stage approach which estimates depth-maps from neighbouring images and then merges them to extract a final surface. Often the depth-maps used for the merging stage will contain outliers due to errors in the matching process. Traditional systems exploit redundancy in the image sequence (the surface is seen in many views), in order to make the final surface estimate robust to these outliers. In the case of sparse data sets there is often insufficient redundancy and thus performance degrades as the number of images decreases. In order to improve performance in these circumstances it is necessary to remove the outliers from the depth-maps. We identify the two main sources of outliers in a top performing algorithm: (1) spurious matches due to repeated texture and (2) matching failure due to occlusion, distortion and lack of texture. We propose two contributions to tackle these failure modes. Firstly, we store multiple depth hypotheses and use a spatial consistency constraint to extract the true depth. Secondly, we allow the algorithm to return an unknownstate when the a true depth estimate cannot be found. By combining these in a discrete label MRF optimisation we are able to obtain high accuracy depth-maps with low numbers of outliers. We evaluate our algorithm in a multi-view stereo framework and find it to confer state-of-the-art performance with the leading techniques, in particular on the standard evaluation sparse data sets. We develop and analyze methods for computing provably optimal maximum a posteriori probability (MAP) configurations for a subclass of Markov random fields defined on graphs with cycles. By decomposing the original distribution into a convex combination of tree-structured distributions, we obtain an upper bound on the optimal value of the original problem (i.e., the log probability of the MAP assignment) in terms of the combined optimal values of the tree problems. We prove that this upper bound is tight if and only if all the tree distributions share an optimal configuration in common. An important implication is that any such shared configuration must also be a MAP configuration for the original distribution. Next we develop two approaches to attempting to obtain tight upper bounds: a) a tree-relaxed linear program (LP), which is derived from the Lagrangian dual of the upper bounds; and b) a tree-reweighted max-product message-passing algorithm that is related to but distinct from the max-product algorithm. In this way, we establish a connection between a certain LP relaxation of the mode-finding problem and a reweighted form of the max-product (min-sum) message-passing algorithm. We introduce a dynamic mechanism for the solution of analytically-tractable substructure in probabilistic programs, using conjugate priors and affine transformations to reduce variance in Monte Carlo estimators. For inference with Sequential Monte Carlo, this automatically yields improvements such as locally-optimal proposals and Rao-Blackwellization. The mechanism maintains a directed graph alongside the running program that evolves dynamically as operations are triggered upon it. Nodes of the graph represent random variables, edges the analytically-tractable relationships between them. Random variables remain in the graph for as long as possible, to be sampled only when they are used by the program in a way that cannot be resolved analytically. In the meantime, they are conditioned on as many observations as possible. We demonstrate the mechanism with a few pedagogical examples, as well as a linear-nonlinear state-space model with simulated data, and an epidemiological model with real data of a dengue outbreak in Micronesia. In all cases one or more variables are automatically marginalized out to significantly reduce variance in estimates of the marginal likelihood, in the final case facilitating a random-weight or pseudo-marginal-type importance sampler for parameter estimation. We have implemented the approach in Anglican and a new probabilistic programming language called Birch. This paper considers constrained lscr1 minimization methods in a unified framework for the recovery of high-dimensional sparse signals in three settings: noiseless, bounded error, and Gaussian noise. Both lscr1 minimization with an lscrinfin constraint (Dantzig selector) and lscr1 minimization under an llscr2 constraint are considered. The results of this paper improve the existing results in the literature by weakening the conditions and tightening the error bounds. The improvement on the conditions shows that signals with larger support can be recovered accurately. In particular, our results illustrate the relationship between lscr1 minimization with an llscr2 constraint and lscr1 minimization with an lscrinfin constraint. This paper also establishes connections between restricted isometry property and the mutual incoherence property. Some results of Candes, Romberg, and Tao (2006), Candes and Tao (2007), and Donoho, Elad, and Temlyakov (2006) are extended. Rapidly-exploring random trees (RRTs) are pop- ular in motion planning because they find solutions efficiently to single-query problems. Optimal RRTs (RRT*s) extend RRTs to the problem of finding the optimal solution, but in doing so asymptotically find the optimal path from the initial state to every state in the planning domain. This behaviour is not only inefficient but also inconsistent with their single-query nature. For problems seeking to minimize path length, the subset of states that can improve a solution can be described by a prolate hyperspheroid. We show that unless this subset is sam- pled directly, the probability of improving a solution becomes arbitrarily small in large worlds or high state dimensions. In this paper, we present an exact method to focus the search by directly sampling this subset. The advantages of the presented sampling technique are demonstrated with a new algorithm, Informed RRT*. This method retains the same probabilistic guarantees on complete- ness and optimality as RRT* while improving the convergence rate and final solution quality. We present the algorithm as a simple modification to RRT* that could be further extended by more advanced path-planning algorithms. We show exper- imentally that it outperforms RRT* in rate of convergence, final solution cost, and ability to find difficult passages while demonstrating less dependence on the state dimension and range of the planning problem. Research on universal and minimax wavelet shrinkage and thresholding methods has demonstrated near-ideal estimation performance in various asymptotic frameworks. However, image processing practice has shown that universal thresholding methods are outperformed by simple Bayesian estimators assuming independent wavelet coefficients and heavy-tailed priors such as generalized Gaussian distributions (GGDs). In this paper, we investigate various connections between shrinkage methods and maximum a posteriori (MAP) estimation using such priors. In particular, we state a simple condition under which MAP estimates are sparse. We also introduce a new family of complexity priors based upon Rissanen's universal prior on integers. One particular estimator in this class outperforms conventional estimators based on earlier applications of the minimum description length (MDL) principle. We develop analytical expressions for the shrinkage rules implied by GGD and complexity priors. This allows us to show the equivalence between universal hard thresholding, MAP estimation using a very heavy-tailed GGD, and MDL estimation using one of the new complexity priors. Theoretical analysis supported by numerous practical experiments shows the robustness of some of these estimates against mis-specifications of the prior-a basic concern in image processing applications.