TY - JOUR
AB - We study the problem of recovering an unknown signal š„š„ given measurements obtained from a generalized linear model with a Gaussian sensing matrix. Two popular solutions are based on a linear estimator š„š„^L and a spectral estimator š„š„^s. The former is a data-dependent linear combination of the columns of the measurement matrix, and its analysis is quite simple. The latter is the principal eigenvector of a data-dependent matrix, and a recent line of work has studied its performance. In this paper, we show how to optimally combine š„š„^L and š„š„^s. At the heart of our analysis is the exact characterization of the empirical joint distribution of (š„š„,š„š„^L,š„š„^s) in the high-dimensional limit. This allows us to compute the Bayes-optimal combination of š„š„^L and š„š„^s, given the limiting distribution of the signal š„š„. When the distribution of the signal is Gaussian, then the Bayes-optimal combination has the form šš„š„^L+š„š„^s and we derive the optimal combination coefficient. In order to establish the limiting distribution of (š„š„,š„š„^L,š„š„^s), we design and analyze an approximate message passing algorithm whose iterates give š„š„^L and approach š„š„^s. Numerical simulations demonstrate the improvement of the proposed combination with respect to the two methods considered separately.
AU - Mondelli, Marco
AU - Thrampoulidis, Christos
AU - Venkataramanan, Ramji
ID - 10211
JF - Foundations of Computational Mathematics
KW - Applied Mathematics
KW - Computational Theory and Mathematics
KW - Computational Mathematics
KW - Analysis
SN - 1615-3375
TI - Optimal combination of linear and spectral estimators for generalized linear models
ER -
TY - CONF
AB - This paper characterizes the latency of the simplified successive-cancellation (SSC) decoding scheme for polar codes under hardware resource constraints. In particular, when the number of processing elements P that can perform SSC decoding operations in parallel is limited, as is the case in practice, the latency of SSC decoding is O(N1ā1 Ī¼+NPlog2log2NP), where N is the block length of the code and Ī¼ is the scaling exponent of polar codes for the channel. Three direct consequences of this bound are presented. First, in a fully-parallel implementation where P=N2 , the latency of SSC decoding is O(N1ā1/Ī¼) , which is sublinear in the block length. This recovers a result from an earlier work. Second, in a fully-serial implementation where P=1 , the latency of SSC decoding scales as O(Nlog2log2N) . The multiplicative constant is also calculated: we show that the latency of SSC decoding when P=1 is given by (2+o(1))Nlog2log2N . Third, in a semi-parallel implementation, the smallest P that gives the same latency as that of the fully-parallel implementation is P=N1/Ī¼ . The tightness of our bound on SSC decoding latency and the applicability of the foregoing results is validated through extensive simulations.
AU - Hashemi, Seyyed Ali
AU - Mondelli, Marco
AU - Fazeli, Arman
AU - Vardy, Alexander
AU - Cioffi, John
AU - Goldsmith, Andrea
ID - 10053
SN - 2157-8095
T2 - 2021 IEEE International Symposium on Information Theory
TI - Parallelism versus latency in simplified successive-cancellation decoding of polar codes
ER -
TY - CONF
AB - This work analyzes the latency of the simplified successive cancellation (SSC) decoding scheme for polar codes proposed by Alamdar-Yazdi and Kschischang. It is shown that, unlike conventional successive cancellation decoding, where latency is linear in the block length, the latency of SSC decoding is sublinear. More specifically, the latency of SSC decoding is O(N 1ā1/Āµ ), where N is the block length and Āµ is the scaling exponent of the channel, which captures the speed of convergence of the rate to capacity. Numerical results demonstrate the tightness of the bound and show that most of the latency reduction arises from the parallel decoding of subcodes of rate 0 and 1.
AU - Mondelli, Marco
AU - Hashemi, Seyyed Ali
AU - Cioffi, John
AU - Goldsmith, Andrea
ID - 8536
SN - 21578095
T2 - IEEE International Symposium on Information Theory - Proceedings
TI - Simplified successive cancellation decoding of polar codes has sublinear latency
VL - 2020-June
ER -
TY - JOUR
AB - Fitting a function by using linear combinations of a large number N of `simple' components is one of the most fruitful ideas in statistical learning. This idea lies at the core of a variety of methods, from two-layer neural networks to kernel regression, to boosting. In general, the resulting risk minimization problem is non-convex and is solved by gradient descent or its variants. Unfortunately, little is known about global convergence properties of these approaches.
Here we consider the problem of learning a concave function f on a compact convex domain Ī©āād, using linear combinations of `bump-like' components (neurons). The parameters to be fitted are the centers of N bumps, and the resulting empirical risk minimization problem is highly non-convex. We prove that, in the limit in which the number of neurons diverges, the evolution of gradient descent converges to a Wasserstein gradient flow in the space of probability distributions over Ī©. Further, when the bump width Ī“ tends to 0, this gradient flow has a limit which is a viscous porous medium equation. Remarkably, the cost function optimized by this gradient flow exhibits a special property known as displacement convexity, which implies exponential convergence rates for Nāā, Ī“ā0. Surprisingly, this asymptotic theory appears to capture well the behavior for moderate values of Ī“,N. Explaining this phenomenon, and understanding the dependence on Ī“,N in a quantitative manner remains an outstanding challenge.
AU - Javanmard, Adel
AU - Mondelli, Marco
AU - Montanari, Andrea
ID - 6748
IS - 6
JF - Annals of Statistics
TI - Analysis of a two-layer neural network via displacement convexity
VL - 48
ER -
TY - JOUR
AB - We prove that, for the binary erasure channel (BEC), the polar-coding paradigm gives rise to codes that not only approach the Shannon limit but do so under the best possible scaling of their block length as a function of the gap to capacity. This result exhibits the first known family of binary codes that attain both optimal scaling and quasi-linear complexity of encoding and decoding. Our proof is based on the construction and analysis of binary polar codes with large kernels. When communicating reliably at rates within Īµ>0 of capacity, the code length n often scales as O(1/ĪµĪ¼), where the constant Ī¼ is called the scaling exponent. It is known that the optimal scaling exponent is Ī¼=2, and it is achieved by random linear codes. The scaling exponent of conventional polar codes (based on the 2Ć2 kernel) on the BEC is Ī¼=3.63. This falls far short of the optimal scaling guaranteed by random codes. Our main contribution is a rigorous proof of the following result: for the BEC, there exist āĆā binary kernels, such that polar codes constructed from these kernels achieve scaling exponent Ī¼(ā) that tends to the optimal value of 2 as ā grows. We furthermore characterize precisely how large ā needs to be as a function of the gap between Ī¼(ā) and 2. The resulting binary codes maintain the recursive structure of conventional polar codes, and thereby achieve construction complexity O(n) and encoding/decoding complexity O(nlogn).
AU - Fazeli, Arman
AU - Hassani, Hamed
AU - Mondelli, Marco
AU - Vardy, Alexander
ID - 9002
IS - 9
JF - IEEE Transactions on Information Theory
SN - 0018-9448
TI - Binary linear codes with optimal scaling: Polar codes with large kernels
VL - 67
ER -
TY - CONF
AB - The optimization of multilayer neural networks typically leads to a solution
with zero training error, yet the landscape can exhibit spurious local minima
and the minima can be disconnected. In this paper, we shed light on this
phenomenon: we show that the combination of stochastic gradient descent (SGD)
and over-parameterization makes the landscape of multilayer neural networks
approximately connected and thus more favorable to optimization. More
specifically, we prove that SGD solutions are connected via a piecewise linear
path, and the increase in loss along this path vanishes as the number of
neurons grows large. This result is a consequence of the fact that the
parameters found by SGD are increasingly dropout stable as the network becomes
wider. We show that, if we remove part of the neurons (and suitably rescale the
remaining ones), the change in loss is independent of the total number of
neurons, and it depends only on how many neurons are left. Our results exhibit
a mild dependence on the input dimension: they are dimension-free for two-layer
networks and depend linearly on the dimension for multilayer networks. We
validate our theoretical findings with numerical experiments for different
architectures and classification tasks.
AU - Shevchenko, Alexander
AU - Mondelli, Marco
ID - 9198
T2 - Proceedings of the 37th International Conference on Machine Learning
TI - Landscape connectivity and dropout stability of SGD solutions for over-parameterized neural networks
VL - 119
ER -
TY - CONF
AB - Recent works have shown that gradient descent can find a global minimum for over-parameterized neural networks where the widths of all the hidden layers scale polynomially with N (N being the number of training samples). In this paper, we prove that, for deep networks, a single layer of width N following the input layer suffices to ensure a similar guarantee. In particular, all the remaining layers are allowed to have constant widths, and form a pyramidal topology. We show an application of our result to the widely used LeCunās initialization and obtain an over-parameterization requirement for the single wide layer of order N2.
AU - Nguyen, Quynh
AU - Mondelli, Marco
ID - 9221
T2 - 34th Conference on Neural Information Processing Systems
TI - Global convergence of deep networks with one wide layer followed by pyramidal topology
VL - 33
ER -
TY - JOUR
AB - We consider the primitive relay channel, where the source sends a message to the relay and to the destination, and the relay helps the communication by transmitting an additional message to the destination via a separate channel. Two well-known coding techniques have been introduced for this setting: decode-and-forward and compress-and-forward. In decode-and-forward, the relay completely decodes the message and sends some information to the destination; in compress-and-forward, the relay does not decode, and it sends a compressed version of the received signal to the destination using WynerāZiv coding. In this paper, we present a novel coding paradigm that provides an improved achievable rate for the primitive relay channel. The idea is to combine compress-and-forward and decode-and-forward via a chaining construction. We transmit over pairs of blocks: in the first block, we use compress-and-forward; and, in the second block, we use decode-and-forward. More specifically, in the first block, the relay does not decode, it compresses the received signal via WynerāZiv, and it sends only part of the compression to the destination. In the second block, the relay completely decodes the message, it sends some information to the destination, and it also sends the remaining part of the compression coming from the first block. By doing so, we are able to strictly outperform both compress-and-forward and decode-and-forward. Note that the proposed coding scheme can be implemented with polar codes. As such, it has the typical attractive properties of polar coding schemes, namely, quasi-linear encoding and decoding complexity, and error probability that decays at super-polynomial speed. As a running example, we take into account the special case of the erasure relay channel, and we provide a comparison between the rates achievable by our proposed scheme and the existing upper and lower bounds.
AU - Mondelli, Marco
AU - Hassani, S. Hamed
AU - Urbanke, RĆ¼diger
ID - 7007
IS - 10
JF - Algorithms
SN - 1999-4893
TI - A new coding paradigm for the primitive relay channel
VL - 12
ER -
TY - JOUR
AB - Polar codes have gained extensive attention during the past few years and recently they have been selected for the next generation of wireless communications standards (5G). Successive-cancellation-based (SC-based) decoders, such as SC list (SCL) and SC flip (SCF), provide a reasonable error performance for polar codes at the cost of low decoding speed. Fast SC-based decoders, such as Fast-SSC, Fast-SSCL, and Fast-SSCF, identify the special constituent codes in a polar code graph off-line, produce a list of operations, store the list in memory, and feed the list to the decoder to decode the constituent codes in order efficiently, thus increasing the decoding speed. However, the list of operations is dependent on the code rate and as the rate changes, a new list is produced, making fast SC-based decoders not rate-flexible. In this paper, we propose a completely rate-flexible fast SC-based decoder by creating the list of operations directly in hardware, with low implementation complexity. We further propose a hardware architecture implementing the proposed method and show that the area occupation of the rate-flexible fast SC-based decoder in this paper is only 38% of the total area of the memory-based base-line decoder when 5G code rates are supported.
AU - Hashemi, Seyyed Ali
AU - Condo, Carlo
AU - Mondelli, Marco
AU - Gross, Warren J
ID - 6750
IS - 22
JF - IEEE Transactions on Signal Processing
SN - 1053587X
TI - Rate-flexible fast polar decoders
VL - 67
ER -