In this task, you are given the abstract of a research paper. Your task is to generate a summary of this abstract. Your summary should not be very short, but it's better if it's not more than 30 words.

Input: Consider Input: The goal of compressed sensing is to learn a structured signal $x$
from a limited number of noisy linear measurements $y pprox Ax$. In
traditional compressed sensing, ''structure'' is represented by
sparsity in some known basis. Inspired by the success of deep
learning in modeling images, recent work starting with~\cite{BDJP17}
has instead considered structure to come from a generative model
$G: \R^k o \R^n$. We present two results establishing the
difficulty of this latter task, showing that existing bounds are
tight.

First, we provide a lower bound matching the~\cite{BDJP17} upper
bound for compressed sensing from $L$-Lipschitz generative models
$G$. In particular, there exists such a function that requires
roughly $\Omega(k \log L)$ linear measurements for sparse recovery
to be possible. This holds even for the more relaxed goal of
\emph{nonuniform} recovery.

Second, we show that generative models generalize sparsity as a
representation of structure. In particular, we construct a
ReLU-based neural network $G: \R^{2k} o \R^n$ with $O(1)$ layers
and $O(kn)$ activations per layer, such that the range of $G$
contains all $k$-sparse vectors.

Output: Lower bound for compressed sensing w/ generative models that matches known upper bounds

Input: Consider Input: Momentary fluctuations in attention (perceptual accuracy) correlate with neural activity fluctuations in primate visual areas. Yet, the link between such momentary neural fluctuations and attention state remains to be shown in the human brain. We investigate this link using a real-time cognitive brain machine interface (cBMI) based on steady state visually evoked potentials (SSVEPs): occipital EEG potentials evoked by rhythmically flashing stimuli. Tracking momentary fluctuations in SSVEP power, in real-time, we presented stimuli time-locked to when this power reached (predetermined) high or low thresholds. We observed a significant increase in discrimination accuracy (d') when stimuli were triggered during high (versus low) SSVEP power epochs, at the location cued for attention. Our results indicate a direct link between attention's effects on perceptual accuracy and and neural gain in EEG-SSVEP power, in the human brain.

Output: With a cognitive brain-machine interface, we show a direct link between attentional effects on perceptual accuracy and neural gain in EEG-SSVEP power, in the human brain.

Input: Consider Input: Hierarchical Bayesian methods have the potential to unify many related tasks (e.g. k-shot classification, conditional, and unconditional generation) by framing each as inference within a single generative model. We show that existing approaches for learning such models can fail on expressive generative networks such as PixelCNNs, by describing the global distribution with little reliance on latent variables. To address this, we develop a modification of the Variational Autoencoder in which encoded observations are decoded to new elements from the same class; the result, which we call a Variational Homoencoder (VHE), may be understood as training a hierarchical latent variable model which better utilises latent variables in these cases. Using this framework enables us to train a hierarchical PixelCNN for the Omniglot dataset, outperforming all existing models on test set likelihood. With a single model we achieve both strong one-shot generation and near human-level classification, competitive with state-of-the-art discriminative classifiers. The VHE objective extends naturally to richer dataset structures such as factorial or hierarchical categories, as we illustrate by training models to separate character content from simple variations in drawing style, and to generalise the style of an alphabet to new characters.
Output: Technique for learning deep generative models with shared latent variables, applied to Omniglot with a PixelCNN decoder.