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The block coordinate descent (BCD) method is widely used for minimizing a continuous function$f$of several block variables. At each iteration of this method, a single block of variables is optimized, while the remaining variables are held fixed. To ensure the convergence of the BCD method, the subproblem of each block variable needs to be solved to its unique global optimal. Unfortunately, this requirement is often too restrictive for many practical scenarios. In this paper, we study an alternative inexact BCD approach which updates the variable blocks by successively minimizing a sequence of approximations of$f$which are either locally tight upper bounds of$f$or strictly convex local approximations of$f$ . The main contributions of this work include the characterizations of the convergence conditions for a fairly wide class of such methods, especially for the cases where the objective functions are either nondifferentiable or nonconvex. Our results unify and extend the existing convergence results for many classical algorithms such as the BCD method, the difference of convex functions (DC) method, the expectation maximization (EM) algorithm, as well as the block forward-backward splitting algorithm, all of which are popular for large scale optimization problems involving big data. [PUBLICATION ABSTRACT]

In the paper, the peak solutions of a class equation is studied, the peak solutions are the maximal and minimal solutions. There is an iterative algorithm given for the solutions of the class equation. First, the existence of the peak solutions of the class equations is obtained. Second, when the peak solutions exist, an iterative algorithm is established to converge to the peak solutions of the class equation. By an upper bound and a lower bound of the solutions of the equation solution as the initial matrix, the iterative algorithm of the paper converges to the peak solutions of the class equation. The convergence problem of the algorithm is proved by the mathematical induction in the paper. The above results are verified by the examples.

Identifying all frequent high utility occupancy itemsets (FHUOIs) in a quantitative transaction dataset is a new trend in data mining. By combining both factors of frequency and utility occupancy, these patterns are more suitable for several applications in the real world. These patterns not only reflect the interests of most users but also contribute a high proportion of the utility in supporting transactions. Nonetheless, the set of all discovered FHUOIs may be very large, especially for large and dense datasets or for low values of predefined minimum thresholds. For this reason, it is often quite challenging for users to analyze and use the obtained patterns. To address this issue, this paper proposes two novel algorithms named MaxCloFHUOIM and CloFHUOIM to extract compact representations of FHUOIs. The former is designed to simultaneously mine two concise representations of FHUOIs that consist of all closed FHUOIs and all maximal FHUOIs, whereas the latter only discovers the closed FHUOIs, which provide a lossless summary of all FHUOIs. The proposed algorithms rely on a novel weak upper bound on utility occupancy, to reduce the search space by quickly pruning itemsets with low utility occupancy. Especially, the algorithms integrate two new efficient strategies to prune non-closed FHUOI candidate branches early in the prefix search tree. Results from an in-depth experimental evaluation conducted on several benchmark real-life and synthetic quantitative datasets demonstrate that MaxCloFHUOIM and CloFHUOIM have excellent performance in terms of runtime, memory usage, and scalability. In particular, the proposed algorithms are up to two orders of magnitude faster than a baseline algorithm.

The assessment of uncertainty prediction has become a necessity for most modeling studies within the hydrology community. This paper addresses uncertainty analysis on a novel hybrid double feedforward neural network (HDFNN) model for generating the sediment load prediction interval (PI). By using the Lower Upper Bound Estimation (LUBE) method, the lower and upper bounds are directly generated as outputs of neural network based models. Coverage Width-based Criterion (CWC) is employed as an objective function for searching high quality PIs. The LUBE-based model is then applied to estimate sediment loads of Muddy Creek in Montana of USA. Results demonstrate the suitability of HDFNN-LUBE model in producing PI in both 90% and 95% confidence levels (CL). It is capable of generating appropriate lower bounds of PIs with narrow intervals. Partitioning analysis reveals consistently excellent performances of HDFNN model in constructing PI in terms of low, medium and high loads. These results therefore verify the reliability and potentiality of the HDFNN model for sediment load estimation with uncertainty. LUBE shows its efficiency in uncertainty prediction as well, which could be used to quantify total uncertainty of data-driven models.

High utility sequence mining is a popular data mining task, which aims at finding sequences having a high utility (importance) in a quantitative sequence database. Though it has several applications, state-of-the-art algorithms have one or more of the following limitations: (1) they rely on a utility function that tends to be biased toward finding long patterns, (2) some algorithms do take pattern length into account using an average-utility function but they adopt an optimistic perspective that can be risky or misleading for some applications, (3) they do not let the user specify additional constraints on patterns to be found. To address these three limitations, this paper defines a novel task of mining frequent high minimum average-utility sequences (FHAUS) with constraints in a quantitative sequence database. This task has the following benefits. First, it uses the average-utility au function based on the minimum utility, which takes the length of a pattern into account to calculate its utility. This helps finding short patterns missed by traditional algorithms and it is based on more safe pessimistic utility calculations. Second, the user can specify a set of monotonic and anti-monotonic constraints C on patterns to filter irrelevant patterns and improve the performance of the mining process. To efficiently find all FHAUSs with constraints, this paper first proposes some novel upper bounds (UBs) and weak upper bounds (WUBs) on the average-utility, which satisfy downward-closure (DC) properties or DC-like properties. Then, to effectively reduce the search space, the paper designs novel width pruning, depth pruning, reducing, and tightening strategies based on the proposed bounds. These proposed novel theoretical results are integrated into an algorithm named C-FHAUSPM (Constrained Frequent High minimum Average-Utility Sequential Pattern Mining) for efficiently discovering all FHAUSs with constraints. Results from extensive experiments on both real-life and synthetic quantitative sequence databases show that C-FHAUSPM is highly efficient in terms of runtime and memory usage.

Recently, the method called tensor completion by parallel matrix factorization via tensor train (TMac-TT) has achieved promising performance on estimating the missing information. TMac-TT, which borrows k e t a u g m e n t a t i o n to transform a lower-order tensor into a higher-order tensor, suffers from serious block-artifacts. To tackle this issue, we build an optimization model combining low-rank matrix factorization based on tensor train (TT) rank and the total variation to retain the strength of TT rank and alleviate block-artifacts. We develop a block successive upper-bound minimization algorithm to solve the proposed model. Under some mild conditions, we theoretically prove that the proposed algorithm converges to the coordinatewise minimizers. Extensive numerical experiments illustrate the superiority of the proposed method over several existing state-of-the-art methods qualitatively and quantitatively.

The identification of both closed frequent high average utility itemsets (CFHAUIs) and generators of frequent high average utility itemsets (GFHAUIs) has substantial significance because they play an essential and concise role in representing frequent high average utility itemsets (FHAUIs). These concise summaries offer a compact yet crucial overview that can be much smaller. In addition, they allow the generation of non-redundant high average utility association rules, a crucial factor for decision-makers to consider. However, difficulty arises from the complexity of discovering these representations, primarily because the average utility function does not satisfy both monotonic and anti-monotonic properties within each equivalence class, that is for itemsets sharing the same subset of transactions. To tackle this challenge, this paper proposes an innovative method for efficiently extracting CFHAUIs and GFHAUIs. This approach introduces novel bounds on the average utility, including a weak lower bound called wlbau and a lower bound named auvlb. Efficient pruning strategies are also designed with the aim of early elimination of non-closed and/or non-generator FHAUIs based on the wlbau and auvlb bounds, leading to quicker execution and lower memory consumption. Additionally, the paper introduces a novel algorithm, CG-FHAUI, designed to concurrently discover both GFHAUIs and CFHAUIs. Empirical results highlight the superior performance of the proposed algorithm in terms of runtime, memory usage, and scalability when compared to a baseline algorithm.

This paper studies the optimal time decay to the Oldroyd-B model in R d ( d ≥ 2 ). By appealing to a refined pure energy method, we prove the lower and upper bounds of decay estimates of the global solutions. In particular, the lower bound of decay estimates is established by fully exploiting the structure of this system and by introducing a new combined quantity.

We present a new computable geometry-based upper bound on the capacity of peak-power-limited and band-limited signal over the Additive White Gaussian Noise Channel. The peak limit applies at continuous time. The bound is a function of the volume and shape of the transmitted signal set, namely the body, in the space of Nyquist-rate samples, comprising all of the points the transmitted signal can reach. At a high SNR, the bound is tight, better than previously known upper bounds and, together with a known lower bound, provides the capacity at an asymptotically high SNR. We found, using a numerical evaluation, the high-SNR capacity of signals with the structure used in Cyclic Prefix assisted Frequency Domain Equalization (CP-FDE) and OFDM for sequence length of up to 100 Nyquist intervals, and we present a conjecture that this result is correct for any sequence length and does not depend on the CPA-FDE structure. This paper extends the methodology developed in previous works. The penalty in power efficiency at a high SNR due to the peak power constraint relative to an average power constraint is about 7.5 dB in the low-pass case and about 5.4 dB in the band-pass case.

For a general M -matrix, we construct a specialized matrix to derive monotonically increasing lower bounds and monotonically decreasing upper bounds for its minimum eigenvalue. These results generalize and significantly improve upon existing related findings. Furthermore, we rigorously prove the monotonicity and convergence properties of these bounds. Finally, for a non-defective M -matrix, we propose a smoothing algorithm to compute its minimum eigenvalue, and we validate the effectiveness of the algorithm through numerical examples.