Computational geometry

We give a unified ("basis free") framework for the Descartes method for real root isolation of square-free real polynomials. This framework encompasses the usual Descartes' rule of sign method for polynomials in the power basis as well as its analog in the Bernstein basis. We then give a new bound on the size of the recursion tree in the Descartes method for polynomials with real coefficients. Applied to polynomials A(X) = Eni=0aiXi with integer coefficients |ai|

https://domino.mpi-inf.mpg.de/intranet/ag1/ag1publ.nsf/d8ab60b7b27d8142c12569e30040d645/6e09cb7fbe62ea64c12571b20040a5d4/$file/esy-desctree-issac06-authprep.pdf

Almost tight recursion tree bounds for the Descartes method

Nonrobustness refers to qualitative or catastrophic failures in geometric algorithms arising from numerical errors. Section 45.1 provides background on these problems. Although nonrobustness is already an issue in “purely numerical” computation, the problem is compounded in “geometric computation.” In Section 45.2 we characterize such computations. Researchers trying to create robust geometric software have tried two approaches: making fixed-precision computation robust (Section 45.3), and making the exact approach viable (Section 45.4). Another source of nonrobustness is the phenomenon of degenerate inputs. General methods for treating degenerate inputs are described in Section 45.5. For some problems the exact approach may be expensive or infeasible. To ensure robustness in this setting, a recent extension of exact computation, the so-called “soft exact approach,” has been proposed. This is described in Section 45.6.

Robust Geometric Computation.

https://pure.mpg.de/pubman/item/item_2076012_3/component/file_2214467/1308.4088v2

Computing real roots of real polynomials

On the asymptotic and practical complexity of solving bivariate systems over the reals

This paper presents two results on the complexity of root isolation via Sturm sequences. Both results exploit amortization arguments.

Amortized Bound for Root Isolation via Sturm Sequences

There is a growing interest in numeric-algebraic techniques in the computer algebra community as such techniques can speed up many applications. This paper is concerned with one such approach called Exact Numeric Computation (ENC). The ENC approach to algebraic number computation is based on iterative verified approximations, combined with constructive zero bounds. This paper describes Core 2, the latest version of the Core Library, a package designed for applications such as non-linear computational geometry. The adaptive complexity of ENC combined with filters makes such libraries practical.

Core 2 smoothly integrates our algebraic ENC subsystem with transcendental functions with e-accurate comparisons. This paper describes how the design of Core 2 addresses key software issues such as modularity, extensibility, efficiency in a setting that combines algebraic and transcendental elements. Our redesign preserves the original goals of the Core Library, namely, to provide a simple and natural interface for ENC computation to support rapid prototyping and exploration. We present examples, experimental results, and timings for our new system, released as Core Library 2.0.

/pdf/the-design-of-core-2-a-library-for-exact-numeric-computation-1lwca34g5y.pdf

The design of core 2: a library for exact numeric computation in geometry and algebra

Smale’s notion of an approximate zero of an analytic function f: C →C is extended to take into account the errors incurred in the evaluation of the Newton operator. Call this stronger notion a robust approximate zero.. We develop a corresponding robust point estimate for such zeros: we prove that if z0∈ C satisfies α(f,z0)<0.02 then z0 is a robust approximate zero, with the associated zero z* lying in the closed disc ${\overline B}(z_{0},\frac{0.07}{f,z_{0}}$. Here α(f,z), γ(f,z) are standard functions in point estimates.

Suppose f(z) is an L-bit integer square-free polynomial of degree d. Using our new algorithm, we can compute an n-bit absolute approximation of z*∈IR starting from a bigfloat z0, in time O[dM(n+d2(L+lg d)lg(n+L))], where M(n) is the complexity of multiplying n-bit integers.

Robust approximate zeros

High-utility sequential pattern mining (HUSPM) has emerged as an important topic due to its wide application and considerable popularity. However, due to the combinatorial explosion of the search space when the HUSPM problem encounters a low utility threshold or large-scale data, it may be time-consuming and memory-costly to address the HUSPM problem. Several algorithms have been proposed for addressing this problem, but they still cost a lot in terms of running time and memory usage. In this paper, to further solve this problem efficiently, we design a compact structure called sequence projection (seqPro) and propose an efficient algorithm, namely discovering high-utility sequential patterns with the seqPro structure (HUSP-SP). HUSP-SP utilizes the compact seq-array to store the necessary information in a sequence database. The seqPro structure is designed to efficiently calculate candidate patterns’ utilities and upper bound values. Furthermore, a new upper bound on utility, namely tighter reduced sequence utility (TRSU) and two pruning strategies in search space, are utilized to improve the mining performance of HUSP-SP. Experimental results on both synthetic and real-life datasets show that HUSP-SP can significantly outperform the state-of-the-art algorithms in terms of running time, memory usage, search space pruning efficiency, and scalability.

https://dl.acm.org/doi/pdf/10.1145/3597935

HUSP-SP: Faster Utility Mining on Sequence Data

Contemporary news reporting increasingly features multimedia content, motivating research on multimedia event extraction. However, the task lacks annotated multimodal training data and artificially generated training data suffer from the distribution shift from the real-world data. In this paper, we propose Cross-modality Augmented Multimedia Event Learning (CAMEL), which successfully utilizes artificially generated multimodal training data and achieves state-of-the-art performance. Conditioned on unimodal training data, we generate multimodal training data using off-the-shelf image generators like Stable Diffusion and image captioners like BLIP. In order to learn robust features that are effective across domains, we devise an iterative and gradual annealing training strategy. Substantial experiments show that CAMEL surpasses state-of-the-art (SOTA) baselines on the M2E2 benchmark. On multimedia events in particular, we outperform the prior SOTA by 4.2\% F1 on event mention identification and by 9.8\% F1 on argument identification, which demonstrates that CAMEL learns synergistic representations from the two modalities.

Zilin Du

Papers

Amortized Bound for Root Isolation via Sturm Sequences

The design of core 2: a library for exact numeric computation in geometry and algebra

Robust approximate zeros

HUSP-SP: Faster Utility Mining on Sequence Data

Training Multimedia Event Extraction With Generated Images and Captions