Erik Ordentlich

Universal portfolios with side information

We present a sequential investment algorithm, the /spl mu/-weighted universal portfolio with side information, which achieves, to first order in the exponent, the same wealth as the best side-information dependent investment strategy (the best state-constant rebalanced portfolio) determined in hindsight from observed market and side-information outcomes. This is an individual sequence result which shows the difference between the exponential growth wealth of the best state-constant rebalanced portfolio and the universal portfolio with side information is uniformly less than (d/(2n))log (n+1)+(k/n)log 2 for every stock market and side-information sequence and for all time n. Here d=k(m-1) is the number of degrees of freedom in the state-constant rebalanced portfolio with k states of side information and m stocks. The proof of this result establishes a close connection between universal investment and universal data compression.

Universal discrete denoising: known channel

A discrete denoising algorithm estimates the input sequence to a discrete memoryless channel (DMC) based on the observation of the entire output sequence. For the case in which the DMC is known and the quality of the reconstruction is evaluated with a given single-letter fidelity criterion, we propose a discrete denoising algorithm that does not assume knowledge of statistical properties of the input sequence. Yet, the algorithm is universal in the sense of asymptotically performing as well as the optimum denoiser that knows the input sequence distribution, which is only assumed to be stationary. Moreover, the algorithm is universal also in a semi-stochastic setting, in which the input is an individual sequence, and the randomness is due solely to the channel noise. The proposed denoising algorithm is practical, requiring a linear number of register-level operations and sublinear working storage size relative to the input data length.

Embedded block coding in JPEG2000

This paper describes the embedded block coding algorithm at the heart of the JPEG2000 image compression standard. The algorithm achieves excellent compression performance, usually somewhat higher than that of SPIHT with arithmetic coding, but in some cases substantially higher. The algorithm utilizes the same low complexity binary arithmetic coding engine as JBIG2. Together with careful design of the bit-plane coding primitives, this enables comparable execution speed to that observed with the simpler variant of SPIHT without arithmetic coding. The coder offers additional advantages including memory locality, spatial random access and ease of geometric manipulation.

A low-complexity modeling approach for embedded coding of wavelet coefficients

We present a new low-complexity method for modeling and coding the bitplanes of a wavelet-transformed image in a fully embedded fashion. The scheme uses a simple ordering model for embedding, based on the principle that coefficient bits that are likely to reduce the distortion the most should be described first in the encoded bitstream. The ordering model is tied to a conditioning model in a way that deinterleaves the conditioned subsequences of coefficient bits, making them amenable to coding with a very simple, adaptive elementary Golomb (1966) code. The proposed scheme, without relying on zerotrees or arithmetic coding, attains PSNR vs. bit rate performance superior to that of SPIHT, and competitive with its arithmetic coding variant, SPIHT-AC.

The Degrees-of-Freedom of the

The degrees-of-freedom of a K-user Gaussian interference channel (GIC) has been defined to be the multiple of (1/2)log 2 P at which the maximum sum of achievable rates grows with increasing power P. In this paper, we establish that the degrees-of-freedom of three or more user, real, scalar GICs, viewed as a function of the channel coefficients, is discontinuous at points where all of the coefficients are nonzero rational numbers. More specifically, for all K > 2, we find a class of K-user GICs that is dense in the GIC parameter space for which K/2 degrees-of-freedom are exactly achievable, and we show that the degrees-of-freedom for any GIC with nonzero rational coefficients is strictly smaller than K/2. These results are proved using new connections with number theory and additive combinatorics.

K

The degrees-of-freedom of a K-user Gaussian interference channel (GIFC) has been defined to be the multiple of (1/2) log2 P at which the maximum sum of achievable rates grows with increasing P. In this paper, we establish that the degrees-of-freedom of three or more user, real, scalar GIFCs, viewed as a function of the channel coefficients, is discontinuous at points where all of the coefficients are non-zero rational numbers. More specifically, for all K ≫ 2, we find a class of K-user GIFCs that is dense in the GIFC parameter space for which K/2 degrees-of-freedom are exactly achievable, and we show that the degrees-of-freedom for any GIFC with non-zero rational coefficients is strictly smaller than K/2. These results are proved using new connections with number theory and additive combinatorics.

-User Gaussian Interference Channel Is Discontinuous at Rational Channel Coefficients

This paper describes the embedded block coding algorithm at the heart of the JPEG 2000 image compression standard. The paper discusses key considerations which led to the development and adoption of this algorithm, and also investigates performance and complexity issues. The JPEG 2000 coding system achieves excellent compression performance, somewhat higher (and, in some cases, substantially higher) than that of SPIHT with arithmetic coding, a popular benchmark for comparison The algorithm utilizes the same low complexity binary arithmetic coding engine as JBIG2. Together with careful design of the bit-plane coding primitives, this enables comparable execution speed to that observed with the simpler variant of SPIHT without arithmetic coding. The coder offers additional advantages including memory locality, spatial random access and ease of geometric manipulation.

On the Degrees-of-Freedom of the K-user Gaussian interference channel

For a market with m assets consider the minimum, over all possible sequences of asset prices through time n, of the ratio of the final wealth of a nonanticipating investment strategy to the wealth obtained by the best constant rebalanced portfolio computed in hindsight for that price sequence. We show that the maximum value of this ratio over all nonanticipating investment strategies is Vn[Σ2-nH(n1/n,...,nm/n)(n1!...nm!))] -1 where H(.) is the Shannon entropy, and we specify a strategy achieving it. The optimal ratio Vn is shown to decrease only polynomially in n, indicating that the rate of return of the optimal strategy converges uniformly to that of the best constant rebalanced portfolio determined with full hindsight. We also relate this result to the pricing of a new derivative security which might be called the hindsight allocation option.

Embedded block coding in JPEG 2000

We present an extension of the discrete universal denoiser DUDE, specialized for the denoising of grayscale images. The original DUDE is a low-complexity algorithm aimed at recovering discrete sequences corrupted by discrete memoryless noise of known statistical characteristics. It is universal, in the sense of asymptotically achieving, without access to any information on the statistics of the clean sequence, the same performance as the best denoiser that does have access to such information. The DUDE, however, is not effective on grayscale images of practical size. The difficulty lies in the fact that one of the DUDEs key components is the determination of conditional empirical probability distributions of image samples, given the sample values in their neighborhood. When the alphabet is relatively large (as is the case with grayscale images), even for a small-sized neighborhood, the required distributions would be estimated from a large collection of sparse statistics, resulting in poor estimates that would not enable effective denoising. The present work enhances the basic DUDE scheme by incorporating statistical modeling tools that have proven successful in addressing similar issues in lossless image compression. Instantiations of the enhanced framework, which is referred to as iDUDE, are described for examples of additive and nonadditive noise. The resulting denoisers significantly surpass the state of the art in the case of salt and pepper (S&P) and -ary symmetric noise, and perform well for Gaussian noise.

The Cost of Achieving the Best Portfolio in Hindsight

This paper describes a discrete universal denoiser for two dimensional data and also presents an experimental results of its application to noisy binary images. A discrete universal denoiser (DUDE) is introduced for recovering a signal with finite-valued components corrupted by finite-valued, uncorrelated noise. The DUDE is asymptotically optimal and universal, in the sense of asymptotically achieving, without access to any information on the statistics of the clean signal, the same performance as the best denoiser that does have access to such information. It is also practical, and can be implemented in low complexity.