63 research outputs found
On the Existence of Optimal Exact-Repair MDS Codes for Distributed Storage
The high repair cost of (n,k) Maximum Distance Separable (MDS) erasure codes
has recently motivated a new class of codes, called Regenerating Codes, that
optimally trade off storage cost for repair bandwidth. In this paper, we
address bandwidth-optimal (n,k,d) Exact-Repair MDS codes, which allow for any
failed node to be repaired exactly with access to arbitrary d survivor nodes,
where k<=d<=n-1. We show the existence of Exact-Repair MDS codes that achieve
minimum repair bandwidth (matching the cutset lower bound) for arbitrary
admissible (n,k,d), i.e., k<n and k<=d<=n-1. Our approach is based on
interference alignment techniques and uses vector linear codes which allow to
split symbols into arbitrarily small subsymbols.Comment: 20 pages, 6 figure
Exact Regeneration Codes for Distributed Storage Repair Using Interference Alignment
The high repair cost of (n,k) Maximum Distance Separable (MDS) erasure codes
has recently motivated a new class of codes, called Regenerating Codes, that
optimally trade off storage cost for repair bandwidth. On one end of this
spectrum of Regenerating Codes are Minimum Storage Regenerating (MSR) codes
that can match the minimum storage cost of MDS codes while also significantly
reducing repair bandwidth. In this paper, we describe Exact-MSR codes which
allow for any failed nodes (whether they are systematic or parity nodes) to be
regenerated exactly rather than only functionally or information-equivalently.
We show that Exact-MSR codes come with no loss of optimality with respect to
random-network-coding based MSR codes (matching the cutset-based lower bound on
repair bandwidth) for the cases of: (a) k/n <= 1/2; and (b) k <= 3. Our
constructive approach is based on interference alignment techniques, and,
unlike the previous class of random-network-coding based approaches, we provide
explicit and deterministic coding schemes that require a finite-field size of
at most 2(n-k).Comment: to be submitted to IEEE Transactions on Information Theor
Degrees of Freedom of Uplink-Downlink Multiantenna Cellular Networks
An uplink-downlink two-cell cellular network is studied in which the first
base station (BS) with antennas receives independent messages from its
serving users, while the second BS with antennas transmits
independent messages to its serving users. That is, the first and second
cells operate as uplink and downlink, respectively. Each user is assumed to
have a single antenna. Under this uplink-downlink setting, the sum degrees of
freedom (DoF) is completely characterized as the minimum of
,
, , and , where denotes
. The result demonstrates that, for a broad class of network
configurations, operating one of the two cells as uplink and the other cell as
downlink can strictly improve the sum DoF compared to the conventional uplink
or downlink operation, in which both cells operate as either uplink or
downlink. The DoF gain from such uplink-downlink operation is further shown to
be achievable for heterogeneous cellular networks having hotspots and with
delayed channel state information.Comment: 22 pages, 11 figures, in revision for IEEE Transactions on
Information Theor
Convex Optimization for Machine Learning
This book covers an introduction to convex optimization, one of the powerful and tractable optimization problems that can be efficiently solved on a computer. The goal of the book is to
help develop a sense of what convex optimization is, and how it can be used in a widening array of practical contexts with a particular emphasis on machine learning.
The first part of the book covers core concepts of convex sets, convex functions, and related basic definitions that serve understanding convex optimization and its corresponding models. The second part deals with one very useful theory, called duality, which enables us to: (1) gain algorithmic insights; and (2) obtain an approximate solution to non-convex optimization problems which are often difficult to solve. The last part focuses on modern applications in machine learning and deep learning.
A defining feature of this book is that it succinctly relates the “story” of how convex optimization plays a role, via historical examples and trending machine learning applications. Another key feature is that it includes programming implementation of a variety of machine learning algorithms inspired by optimization fundamentals, together with a brief tutorial of the used programming tools. The implementation is based on Python, CVXPY, and TensorFlow.
This book does not follow a traditional textbook-style organization, but is streamlined via a series of lecture notes that are intimately related, centered around coherent themes and concepts. It serves as a textbook mainly for a senior-level undergraduate course, yet is also suitable for a first-year graduate course. Readers benefit from having a good background in linear algebra, some exposure to probability, and basic familiarity with Python
Computation in Multicast Networks: Function Alignment and Converse Theorems
The classical problem in network coding theory considers communication over
multicast networks. Multiple transmitters send independent messages to multiple
receivers which decode the same set of messages. In this work, computation over
multicast networks is considered: each receiver decodes an identical function
of the original messages. For a countably infinite class of two-transmitter
two-receiver single-hop linear deterministic networks, the computing capacity
is characterized for a linear function (modulo-2 sum) of Bernoulli sources.
Inspired by the geometric concept of interference alignment in networks, a new
achievable coding scheme called function alignment is introduced. A new
converse theorem is established that is tighter than cut-set based and
genie-aided bounds. Computation (vs. communication) over multicast networks
requires additional analysis to account for multiple receivers sharing a
network's computational resources. We also develop a network decomposition
theorem which identifies elementary parallel subnetworks that can constitute an
original network without loss of optimality. The decomposition theorem provides
a conceptually-simpler algebraic proof of achievability that generalizes to
-transmitter -receiver networks.Comment: to appear in the IEEE Transactions on Information Theor
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