5 research outputs found
Multitaper estimation on arbitrary domains
Multitaper estimators have enjoyed significant success in estimating spectral
densities from finite samples using as tapers Slepian functions defined on the
acquisition domain. Unfortunately, the numerical calculation of these Slepian
tapers is only tractable for certain symmetric domains, such as rectangles or
disks. In addition, no performance bounds are currently available for the mean
squared error of the spectral density estimate. This situation is inadequate
for applications such as cryo-electron microscopy, where noise models must be
estimated from irregular domains with small sample sizes. We show that the
multitaper estimator only depends on the linear space spanned by the tapers. As
a result, Slepian tapers may be replaced by proxy tapers spanning the same
subspace (validating the common practice of using partially converged solutions
to the Slepian eigenproblem as tapers). These proxies may consequently be
calculated using standard numerical algorithms for block diagonalization. We
also prove a set of performance bounds for multitaper estimators on arbitrary
domains. The method is demonstrated on synthetic and experimental datasets from
cryo-electron microscopy, where it reduces mean squared error by a factor of
two or more compared to traditional methods.Comment: 28 pages, 11 figure
Learning Invariant Representations under General Interventions on the Response
It has become increasingly common nowadays to collect observations of feature
and response pairs from different environments. As a consequence, one has to
apply learned predictors to data with a different distribution due to
distribution shifts. One principled approach is to adopt the structural causal
models to describe training and test models, following the invariance principle
which says that the conditional distribution of the response given its
predictors remains the same across environments. However, this principle might
be violated in practical settings when the response is intervened. A natural
question is whether it is still possible to identify other forms of invariance
to facilitate prediction in unseen environments. To shed light on this
challenging scenario, we focus on linear structural causal models (SCMs) and
introduce invariant matching property (IMP), an explicit relation to capture
interventions through an additional feature, leading to an alternative form of
invariance that enables a unified treatment of general interventions on the
response as well as the predictors. We analyze the asymptotic generalization
errors of our method under both the discrete and continuous environment
settings, where the continuous case is handled by relating it to the
semiparametric varying coefficient models. We present algorithms that show
competitive performance compared to existing methods over various experimental
settings including a COVID dataset.Comment: Accepted to the IEEE Journal on Selected Areas in Information Theory.
Special Issue: Causality: Fundamental Limits and Application