A Combinatorial Formula for Orthogonal Idempotents in the 00-Hecke Algebra of the Symmetric Group

Building on the work of P.N. Norton, we give combinatorial formulae for two maximal decompositions of the identity into orthogonal idempotents in the $0$-Hecke algebra of the symmetric group, $\mathbb{C}H_0(S_N)$. This construction is compatible with the branching from $S_{N-1}$ to $S_{N}$.Comment: 25 pages, 2 figure

Excursions into Algebra and Combinatorics at q=0q=0

We explore combinatorics associated with the degenerate Hecke algebra at $q=0$, obtaining a formula for a system of orthogonal idempotents, and also exploring various pattern avoidance results. Generalizing constructions for the 0-Hecke algebra, we explore the representation theory of \JJ-trivial monoids. We then discuss two-tensors of crystal bases for $U_q(\tilde{\mathfrak{sl}_2})$, establishing a complementary result to one of Bandlow, Schilling, and Thi\'ery on affine crystals arising from promotion operators. Finally, we give a computer implementation of Stembridge's local axioms for simply-laced crystal bases.Comment: 92 pages, 13 figures. PHd Dissertation accepted at the University of California on July 15th, 2011. arXiv admin note: text overlap with arXiv:1012.136

On the representation theory of finite J-trivial monoids

In 1979, Norton showed that the representation theory of the 0-Hecke algebra admits a rich combinatorial description. Her constructions rely heavily on some triangularity property of the product, but do not use explicitly that the 0-Hecke algebra is a monoid algebra. The thesis of this paper is that considering the general setting of monoids admitting such a triangularity, namely J-trivial monoids, sheds further light on the topic. This is a step to use representation theory to automatically extract combinatorial structures from (monoid) algebras, often in the form of posets and lattices, both from a theoretical and computational point of view, and with an implementation in Sage. Motivated by ongoing work on related monoids associated to Coxeter systems, and building on well-known results in the semi-group community (such as the description of the simple modules or the radical), we describe how most of the data associated to the representation theory (Cartan matrix, quiver) of the algebra of any J-trivial monoid M can be expressed combinatorially by counting appropriate elements in M itself. As a consequence, this data does not depend on the ground field and can be calculated in O(n^2), if not O(nm), where n=|M| and m is the number of generators. Along the way, we construct a triangular decomposition of the identity into orthogonal idempotents, using the usual M\"obius inversion formula in the semi-simple quotient (a lattice), followed by an algorithmic lifting step. Applying our results to the 0-Hecke algebra (in all finite types), we recover previously known results and additionally provide an explicit labeling of the edges of the quiver. We further explore special classes of J-trivial monoids, and in particular monoids of order preserving regressive functions on a poset, generalizing known results on the monoids of nondecreasing parking functions.Comment: 41 pages; 4 figures; added Section 3.7.4 in version 2; incorporated comments by referee in version

All Thresholds Barred: Direct Estimation of Call Density in Bioacoustic Data

Passive acoustic monitoring (PAM) studies generate thousands of hours of audio, which may be used to monitor specific animal populations, conduct broad biodiversity surveys, detect threats such as poachers, and more. Machine learning classifiers for species identification are increasingly being used to process the vast amount of audio generated by bioacoustic surveys, expediting analysis and increasing the utility of PAM as a management tool. In common practice, a threshold is applied to classifier output scores, and scores above the threshold are aggregated into a detection count. The choice of threshold produces biased counts of vocalizations, which are subject to false positive/negative rates that may vary across subsets of the dataset. In this work, we advocate for directly estimating call density: The proportion of detection windows containing the target vocalization, regardless of classifier score. Our approach targets a desirable ecological estimator and provides a more rigorous grounding for identifying the core problems caused by distribution shifts -- when the defining characteristics of the data distribution change -- and designing strategies to mitigate them. We propose a validation scheme for estimating call density in a body of data and obtain, through Bayesian reasoning, probability distributions of confidence scores for both the positive and negative classes. We use these distributions to predict site-level densities, which may be subject to distribution shifts. We test our proposed methods on a real-world study of Hawaiian birds and provide simulation results leveraging existing fully annotated datasets, demonstrating robustness to variations in call density and classifier model quality.Comment: 14 pages, 6 figures, 3 tables; submitted to Frontiers in Bird Science; Our Hawaiian PAM dataset and classifier scores, as well as annotation information for the three study species, can be found on Zenodo at https://doi.org/10.5281/zenodo.10581530. The fully annotated Powdermill dataset assembled by Chronister et al. that was used in this study is available at https://doi.org/10.1002/ecy.332

In Search for a Generalizable Method for Source Free Domain Adaptation

Source-free domain adaptation (SFDA) is compelling because it allows adapting an off-the-shelf model to a new domain using only unlabelled data. In this work, we apply existing SFDA techniques to a challenging set of naturally-occurring distribution shifts in bioacoustics, which are very different from the ones commonly studied in computer vision. We find existing methods perform differently relative to each other than observed in vision benchmarks, and sometimes perform worse than no adaptation at all. We propose a new simple method which outperforms the existing methods on our new shifts while exhibiting strong performance on a range of vision datasets. Our findings suggest that existing SFDA methods are not as generalizable as previously thought and that considering diverse modalities can be a useful avenue for designing more robust models

BIRB: A Generalization Benchmark for Information Retrieval in Bioacoustics

The ability for a machine learning model to cope with differences in training and deployment conditions--e.g. in the presence of distribution shift or the generalization to new classes altogether--is crucial for real-world use cases. However, most empirical work in this area has focused on the image domain with artificial benchmarks constructed to measure individual aspects of generalization. We present BIRB, a complex benchmark centered on the retrieval of bird vocalizations from passively-recorded datasets given focal recordings from a large citizen science corpus available for training. We propose a baseline system for this collection of tasks using representation learning and a nearest-centroid search. Our thorough empirical evaluation and analysis surfaces open research directions, suggesting that BIRB fills the need for a more realistic and complex benchmark to drive progress on robustness to distribution shifts and generalization of ML models