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To characterize the computational complexity of satisfiability problems for probabilistic and causal reasoning within the Pearl's Causal Hierarchy, arXiv:2305.09508 [cs.AI] introduce a new natural class, named succ-$\exists$R. This class can be viewed as a succinct variant of the well-studied class $\exists$R based on the Existential Theory of the Reals (ETR). Analogously to $\exists$R, succ-$\exists$R is an intermediate class between NEXP and EXPSPACE, the exponential versions of NP and PSPACE. The main contributions of this work are threefold. Firstly, we characterize the class succ-$\exists$R in terms of nondeterministic real RAM machines and develop structural complexity theoretic results for real RAMs, including translation and hierarchy theorems. Notably, we demonstrate the separation of $\exists$R and succ-$\exists$R. Secondly, we examine the complexity of model checking and satisfiability of fragments of existential second-order logic and probabilistic independence logic. We show succ-$\exists$R- completeness of several of these problems, for which the best-known complexity lower and upper bounds were previously NEXP-hardness and EXPSPACE, respectively. Thirdly, while succ-$\exists$R is characterized in terms of ordinary (non-succinct) ETR instances enriched by exponential sums and a mechanism to index exponentially many variables, in this paper, we prove that when only exponential sums are added, the corresponding class $\exists$R^{\Sigma} is contained in PSPACE. We conjecture that this inclusion is strict, as this class is equivalent to adding a VNP-oracle to a polynomial time nondeterministic real RAM. Conversely, the addition of exponential products to ETR, yields PSPACE. Additionally, we study the satisfiability problem for probabilistic reasoning, with the additional requirement of a small model and prove that this problem is complete for $\exists$R^{\Sigma}.

The claw problem is central in the fields of theoretical computer science as well as cryptography. The optimal quantum query complexity of the problem is known to be $\Omega\left(\sqrt{G}+(FG)^{1/3} \right)$ for input functions $f\colon [F]\to Z$ and $g\colon [G]\to Z$. However, the lower bound was proved when the range $Z$ is sufficiently large (i.e., $|{Z}|=\Omega(FG)$). The current paper proves the lower bound holds even for every smaller range $Z$ with $|{Z}|\ge F+G$. This implies that $\Omega\left(\sqrt{G}+(FG)^{1/3} \right)$ is tight for every such range. In addition, the lower bound $\Omega\left(\sqrt{G}+F^{1/3}G^{1/6}M^{1/6}\right)$ is provided for even smaller range $Z=[M]$ with every $M\in [2,F+G]$ by reducing the claw problem for $|{Z}|= F+G$. The proof technique is general enough to apply to any $k$-symmetric property (e.g., the $k$-claw problem), i.e., the Boolean function $\Phi$ on the set of $k$ functions with different-size domains and a common range such that $\Phi$ is invariant under the permutations over each domain and the permutations over the range. More concretely, it generalizes Ambainis's argument [Theory of Computing, 1(1):37-46] to the multiple-function case by using the notion of multisymmetric polynomials.

Researchers are exploring novel computational paradigms such as sparse coding and neuromorphic computing to bridge the efficiency gap between the human brain and conventional computers in complex tasks. A key area of focus is neuromorphic audio processing. While the Locally Competitive Algorithm has emerged as a promising solution for sparse coding, offering potential for real-time and low-power processing on neuromorphic hardware, its applications in neuromorphic speech classification have not been thoroughly studied. The Adaptive Locally Competitive Algorithm builds upon the Locally Competitive Algorithm by dynamically adjusting the modulation parameters of the filter bank to fine-tune the filters' sensitivity. This adaptability enhances lateral inhibition, improving reconstruction quality, sparsity, and convergence time, which is crucial for real-time applications. This paper demonstrates the potential of the Locally Competitive Algorithm and its adaptive variant as robust feature extractors for neuromorphic speech classification. Results show that the Locally Competitive Algorithm achieves better speech classification accuracy at the expense of higher power consumption compared to the LAUSCHER cochlea model used for benchmarking. On the other hand, the Adaptive Locally Competitive Algorithm mitigates this power consumption issue without compromising the accuracy. The dynamic power consumption is reduced to a range of 4 to 13 milliwatts on neuromorphic hardware, three orders of magnitude less than setups using Graphics Processing Units. These findings position the Adaptive Locally Competitive Algorithm as a compelling solution for efficient speech classification systems, promising substantial advancements in balancing speech classification accuracy and power efficiency.

State-space models (SSMs) that utilize linear, time-invariant (LTI) systems are known for their effectiveness in learning long sequences. To achieve state-of-the-art performance, an SSM often needs a specifically designed initialization, and the training of state matrices is on a logarithmic scale with a very small learning rate. To understand these choices from a unified perspective, we view SSMs through the lens of Hankel operator theory. Building upon it, we develop a new parameterization scheme, called HOPE, for LTI systems that utilizes Markov parameters within Hankel operators. Our approach helps improve the initialization and training stability, leading to a more robust parameterization. We efficiently implement these innovations by nonuniformly sampling the transfer functions of LTI systems, and they require fewer parameters compared to canonical SSMs. When benchmarked against HiPPO-initialized models such as S4 and S4D, an SSM parameterized by Hankel operators demonstrates improved performance on Long-Range Arena (LRA) tasks. Moreover, our new parameterization endows the SSM with non-decaying memory within a fixed time window, which is empirically corroborated by a sequential CIFAR-10 task with padded noise.

Surrogate models are used to predict the behavior of complex energy systems that are too expensive to simulate with traditional numerical methods. Our work introduces the use of language descriptions, which we call "system captions" or SysCaps, to interface with such surrogates. We argue that interacting with surrogates through text, particularly natural language, makes these models more accessible for both experts and non-experts. We introduce a lightweight multimodal text and timeseries regression model and a training pipeline that uses large language models (LLMs) to synthesize high-quality captions from simulation metadata. Our experiments on two real-world simulators of buildings and wind farms show that our SysCaps-augmented surrogates have better accuracy on held-out systems than traditional methods while enjoying new generalization abilities, such as handling semantically related descriptions of the same test system. Additional experiments also highlight the potential of SysCaps to unlock language-driven design space exploration and to regularize training through prompt augmentation.

Clinically deployed deep learning-based segmentation models are known to fail on data outside of their training distributions. While clinicians review the segmentations, these models tend to perform well in most instances, which could exacerbate automation bias. Therefore, detecting out-of-distribution images at inference is critical to warn the clinicians that the model likely failed. This work applied the Mahalanobis distance (MD) post hoc to the bottleneck features of four Swin UNETR and nnU-net models that segmented the liver on T1-weighted magnetic resonance imaging and computed tomography. By reducing the dimensions of the bottleneck features with either principal component analysis or uniform manifold approximation and projection, images the models failed on were detected with high performance and minimal computational load. In addition, this work explored a non-parametric alternative to the MD, a k-th nearest neighbors distance (KNN). KNN drastically improved scalability and performance over MD when both were applied to raw and average-pooled bottleneck features.

We propose a theory for matrix completion that goes beyond the low-rank structure commonly considered in the literature and applies to general matrices of low description complexity. Specifically, complexity of the sets of matrices encompassed by the theory is measured in terms of Hausdorff and upper Minkowski dimensions. Our goal is the characterization of the number of linear measurements, with an emphasis on rank-$1$ measurements, needed for the existence of an algorithm that yields reconstruction, either perfect, with probability 1, or with arbitrarily small probability of error, depending on the setup. Concretely, we show that matrices taken from a set $\mathcal{U}$ such that $\mathcal{U}-\mathcal{U}$ has Hausdorff dimension $s$ can be recovered from $k>s$ measurements, and random matrices supported on a set $\mathcal{U}$ of Hausdorff dimension $s$ can be recovered with probability 1 from $k>s$ measurements. What is more, we establish the existence of recovery mappings that are robust against additive perturbations or noise in the measurements. Concretely, we show that there are $\beta$-H\"older continuous mappings recovering matrices taken from a set of upper Minkowski dimension $s$ from $k>2s/(1-\beta)$ measurements and, with arbitrarily small probability of error, random matrices supported on a set of upper Minkowski dimension $s$ from $k>s/(1-\beta)$ measurements. The numerous concrete examples we consider include low-rank matrices, sparse matrices, QR decompositions with sparse R-components, and matrices of fractal nature.

Common practice in modern machine learning involves fitting a large number of parameters relative to the number of observations. These overparameterized models can exhibit surprising generalization behavior, e.g., ``double descent'' in the prediction error curve when plotted against the raw number of model parameters, or another simplistic notion of complexity. In this paper, we revisit model complexity from first principles, by first reinterpreting and then extending the classical statistical concept of (effective) degrees of freedom. Whereas the classical definition is connected to fixed-X prediction error (in which prediction error is defined by averaging over the same, nonrandom covariate points as those used during training), our extension of degrees of freedom is connected to random-X prediction error (in which prediction error is averaged over a new, random sample from the covariate distribution). The random-X setting more naturally embodies modern machine learning problems, where highly complex models, even those complex enough to interpolate the training data, can still lead to desirable generalization performance under appropriate conditions. We demonstrate the utility of our proposed complexity measures through a mix of conceptual arguments, theory, and experiments, and illustrate how they can be used to interpret and compare arbitrary prediction models.

It is very well-known that when the exact line search gradient descent method is applied to a convex quadratic objective, the worst case rate of convergence (among all seed vectors) deteriorates as the condition number of the Hessian of the objective grows. By an elegant analysis by H. Akaike, it is generally believed -- but not proved -- that in the ill-conditioned regime the ROC for almost all initial vectors, and hence also the average ROC, is close to the worst case ROC. We complete Akaike's analysis using the theorem of center and stable manifolds. Our analysis also makes apparent the effect of an intermediate eigenvalue in the Hessian by establishing the following somewhat amusing result: In the absence of an intermediate eigenvalue, the average ROC gets arbitrarily fast -- not slow -- as the Hessian gets increasingly ill-conditioned. We discuss in passing some contemporary applications of exact line search GD to polynomial optimization problems arising from imaging and data sciences.

Object detection typically assumes that training and test data are drawn from an identical distribution, which, however, does not always hold in practice. Such a distribution mismatch will lead to a significant performance drop. In this work, we aim to improve the cross-domain robustness of object detection. We tackle the domain shift on two levels: 1) the image-level shift, such as image style, illumination, etc, and 2) the instance-level shift, such as object appearance, size, etc. We build our approach based on the recent state-of-the-art Faster R-CNN model, and design two domain adaptation components, on image level and instance level, to reduce the domain discrepancy. The two domain adaptation components are based on H-divergence theory, and are implemented by learning a domain classifier in adversarial training manner. The domain classifiers on different levels are further reinforced with a consistency regularization to learn a domain-invariant region proposal network (RPN) in the Faster R-CNN model. We evaluate our newly proposed approach using multiple datasets including Cityscapes, KITTI, SIM10K, etc. The results demonstrate the effectiveness of our proposed approach for robust object detection in various domain shift scenarios.