3 research outputs found
Information without rolling dice
The deterministic notions of capacity and entropy are studied in the context
of communication and storage of information using square-integrable,
bandlimited signals subject to perturbation. The -capacity,
that extends the Kolmogorov -capacity to packing sets of overlap at
most , is introduced and compared to the Shannon capacity. The
functional form of the results indicates that in both Kolmogorov and Shannon's
settings, capacity and entropy grow linearly with the number of degrees of
freedom, but only logarithmically with the signal to noise ratio. This basic
insight transcends the details of the stochastic or deterministic description
of the information-theoretic model. For , the analysis leads to new
bounds on the Kolmogorov -capacity, and to a tight asymptotic
expression of the Kolmogorov -entropy of bandlimited signals. A
deterministic notion of error exponent is introduced. Applications of the
theory are briefly discussed
Time-Limited Toeplitz Operators on Abelian Groups: Applications in Information Theory and Subspace Approximation
Toeplitz operators are fundamental and ubiquitous in signal processing and
information theory as models for linear, time-invariant (LTI) systems. Due to
the fact that any practical system can access only signals of finite duration,
time-limited restrictions of Toeplitz operators are naturally of interest. To
provide a unifying treatment of such systems working on different signal
domains, we consider time-limited Toeplitz operators on locally compact abelian
groups with the aid of the Fourier transform on these groups. In particular, we
survey existing results concerning the relationship between the spectrum of a
time-limited Toeplitz operator and the spectrum of the corresponding
non-time-limited Toeplitz operator. We also develop new results specifically
concerning the eigenvalues of time-frequency limiting operators on locally
compact abelian groups. Applications of our unifying treatment are discussed in
relation to channel capacity and in relation to representation and
approximation of signals
Noiseless Privacy
In this paper, we define noiseless privacy, as a non-stochastic rival to
differential privacy, requiring that the outputs of a mechanism (i.e., function
composition of a privacy-preserving mapping and a query) can attain only a few
values while varying the data of an individual (the logarithm of the number of
the distinct values is bounded by the privacy budget). Therefore, the output of
the mechanism is not fully informative of the data of the individuals in the
dataset. We prove several guarantees for noiselessly-private mechanisms. The
information content of the output about the data of an individual, even if an
adversary knows all the other entries of the private dataset, is bounded by the
privacy budget. The zero-error capacity of memory-less channels using
noiselessly private mechanisms for transmission is upper bounded by the privacy
budget. The performance of a non-stochastic hypothesis-testing adversary is
bounded again by the privacy budget. Finally, assuming that an adversary has
access to a stochastic prior on the dataset, we prove that the estimation error
of the adversary for individual entries of the dataset is lower bounded by a
decreasing function of the privacy budget. In this case, we also show that the
maximal information leakage is bounded by the privacy budget. In addition to
privacy guarantees, we prove that noiselessly-private mechanisms admit
composition theorem and post-processing does not weaken their privacy
guarantees. We prove that quantization operators can ensure noiseless privacy
if the number of quantization levels is appropriately selected based on the
sensitivity of the query and the privacy budget. Finally, we illustrate the
privacy merits of noiseless privacy using multiple datasets in energy and
transport