Digit Set Research Articles

Boosting neural networks.

Boosting is a general method for improving the performance of learning algorithms. A recently proposed boosting algorithm, AdaBoost, has been applied with great success to several benchmark machine learning problems using mainly decision trees as base classifiers. In this article we investigate whether AdaBoost also works as well with neural networks, and we discuss the advantages and drawbacks of different versions of the AdaBoost algorithm. In particular, we compare training methods based on sampling the training set and weighting the cost function. The results suggest that random resampling of the training data is not the main explanation of the success of the improvements brought by AdaBoost. This is in contrast to bagging, which directly aims at reducing variance and for which random resampling is essential to obtain the reduction in generalization error. Our system achieves about 1.4% error on a data set of on-line handwritten digits from more than 200 writers. A boosted multilayer network achieved 1.5% error on the UCI letters and 8.1% error on the UCI satellite data set, which is significantly better than boosted decision trees.

Acalculia: Deficits of operational and non-operational number knowledge

Dyscalculia, like dyslexia, affects some 5% of school-age children but has received much less investigative attention. In two thirds of affected children, dyscalculia is associated with another developmental disorder like dyslexia, attention-deficit disorder, anxiety disorder, visual and spatial disorder, or cultural deprivation. Infants, primates, some birds, and other animals are born with the innate ability, called subitizing, to tell at a glance whether small sets of scattered dots or other items differ by one or more item. This nonverbal approximate number system extends mostly to single digit sets as visual discrimination drops logarithmically to “many” with increasing numerosity (size effect) and crowding (distance effect). Preschoolers need several years and specific teaching to learn verbal names and visual symbols for numbers and school agers to understand their cardinality and ordinality and the invariance of their sequence (arithmetic number line) that enables calculation. This arithmetic linear line differs drastically from the nonlinear approximate number system mental number line that parallels the individual number-tuned neurons in the intraparietal sulcus in monkeys and overlying scalp distribution of discrete functional magnetic resonance imaging activations by number tasks in man. Calculation is a complex skill that activates both visual and spatial and visual and verbal networks. It is less strongly left lateralized than language, with approximate number system activation somewhat more right sided and exact number and arithmetic activation more left sided. Maturation and increasing number skill decrease associated widespread non-numerical brain activations that persist in some individuals with dyscalculia, which has no single, universal neurological cause or underlying mechanism in all affected individuals.

Note on matrices of random numbers.

The statistical and structural characteristics of 13 matrices of random numbers in which both the cells and the entries were randomly chosen are discussed. Each matrix was explored considering row means, standard deviations, and correlations as well as column means, standard deviations, and correlations. A study concerning the sequential arrangement of digits was performed by finding out in tables of random numbers how many times the values 0 to 9 are followed by any other digit. Analyses indicate clear factor structures when factor analyzing correlations of rows and of columns and when examining sequential arrangements, concluding that for a given set of digits it is possible to assert both randomness and nonrandomness depending on how the data are examined.

A VLSI algorithm for computing the Euclidean norm of a 3D vector

A digit-recurrence algorithm for computing the Euclidean norm of a three-dimensional (3D) vector which often appears in 3D computer graphics is proposed. One of the three squarings required for the usual computation is removed and the other two squarings, as well as the two additions, are overlapped with the square rooting. The Euclidean norm is computed by iteration of carry-propagation-free additions, shifts, and multiplications by one digit. Different specific versions of the algorithm are possible, depending on the radix, the redundancy factor of the digit set, and etc. Each version of the algorithm can be implemented as a sequential (folded) circuit or a combinational (unfolded) circuit, which has a regular array structure suitable for VLSI.

Polygonal radix representations of complex numbers

Complex numbers can be represented in positional notation using certain digit sets. In this paper, we present the polygonal representation which uses zero and the n-roots of unity as digits. We give conditions on the base in order that every complex number be representable in such a system. We totally characterize complete polygonal numeration systems in imaginary quadratic fields.

Output loss or rehearsal loop? Output-time versus pronunciation-time limits in immediate recall for forgetting-matched materials.

Forgetting during recall may be one limit on memory span. Output time and accuracy of immediate serial recall using spoken and keypress responses were measured for digit, letter, and word sets approximately matched in phonemic discriminability and in immediate recognition memory. Nonetheless, the materials differed from one another in recall span, in output time during recall, and in pronunciation time (speech rate). Recall output times accounted precisely and completely for the measured memory span for these matched materials. Pronunciation times are correlated with recall output times, but output time gives a slightly better account of recall accuracy. The output time equivalent to the rule that short-term memory span corresponds to the number of items that can be said in about 1.5-2 s is that span corresponds to the number of items that can be recalled in about 4-6 s. Additional variations in span reflect differential item interference.

Output loss or rehearsal loop? Output-time versus pronunciation-time limits in immediate recall for forgetting-matched materials.

Forgetting during recall may be one limit on memory span. Output time and accuracy of immediate serial recall using spoken and keypress responses were measured for digit, letter, and word sets approximately matched in phonemic discriminability and in immediate recognition memory. Nonetheless, the materials differed from one another in recall span, in output time during recall, and in pronunciation time (speech rate). Recall output times accounted precisely and completely for the measured memory span for these matched materials. Pronunciation times are correlated with recall output times, but output time gives a slightly better account of recall accuracy. The output time equivalent to the rule that short-term memory span corresponds to the number of items that can be said in about 1.5-2 s is that span corresponds to the number of items that can be recalled in about 4-6 s. Additional variations in span reflect differential item interference.

New Svoboda-Tung division

The paper presents a general theory for developing new Svoboda-Tung (or simply NST) division algorithms not suffering the drawbacks of the classical Svoboda-Tung (or simply ST) method. NST avoids the drawbacks of ST by proper recoding of the two most significant digits of the residual before selecting the most significant digit of this recoded residual as the quotient digit. NST relies on the divisor being in the range [1, 1+/spl delta/), where /spl delta/ is a positive fraction depending upon: 1) the radix, 2) the signed digit set used to represent the residual, and 3) the recoding conditions of the two most significant digits of the residual. If the operands belong to the IEEE Std range [1, 2), they have to be conveniently prescaled. In that case, NST produces the correct quotient but the final residual is scaled by the same factor as the operands, therefore, NST is not useful in applications where the unsealed residual is necessary. An analysis of NST shows that previously published algorithms can be derived from the general theory proposed in the paper. Moreover, NST reveals a spectrum of new possibilities for the design of alternative division units. For a given radix-b, the number of different algorithms of this kind is b/sup 2//4.

Fuzzy integration of classification results

This paper attempts to solve the problem of integrating classification results in a multiple classifier system using fuzzy reasoning. We present three fuzzy classification result integration algorithms covering all forms of classification output. We describe algorithms to model classification results and to reason about the unique output result from all types of classification results. The proposed algorithms were implemented and tested through experiments on a training and test data set of handwritten digits cropped from U.S. postal mail pieces.

Automatic maps in exotic numeration systems

We generalize the classical notion of ab-automatic sequence for a sequence indexed by the natural numbers. We replace the integers by a semiring and use a numeration system consisting of the powers of a baseb and an appropriate set of digits. For example, we define (−3)-automatic sequences (indexed by the ordinary integers or by the rational integers) and (−1 +i)-automatic sequences (indexed by the Gaussian integers). We show how these new notions are related to the old ones, and we study both the number-theoretic and automata-theoretic properties that permit the replacement of one numeration system by another.

Radix 2 division with over-redundant quotient selection

In this paper we present a new radix 2 division algorithm that uses a recurrence employing simple 3-to-2 digit carry-free adders to perform carry-free addition/subtraction for computing the partial remainders in radix 2 signed-digit form. The quotient digit, during any iteration of the division recursion, is generated from the two most-significant radix 2 digits of the partial remainder and independent of the divisor in over-redundant radix 2 digit form (i.e., with digits which belong to the digit set {-2, -1, 0, +1, +2}). The over-redundant quotient digits are then converted to the conventional radix 2 digits (belonging to the set {-1, 0, +1}) by using a reduction technique. This division algorithm is well suited for IEEE 754 standard operands belonging to the range (1, 2) and is slightly faster than previously proposed radix 2 designs (such as the radix 2 SRT), which do not employ input scaling, since the quotient selection for such algorithms is a function of more than two most-significant radix 2 digits of the partial remainder. In comparison with the designs that employ input scaling, the proposed design although slightly slower saves hardware required for scaling purposes.

Integral self-affine tiles in ℝ n part II: Lattice tilings

Let A be an expanding n × n integer matrix with |det (A)| = m. A standard digit set \({\cal D}\) for A is any complete set of coset representatives for \({\Bbb Z}^n / A ( {\Bbb Z}^n ).\) Associated to a given \({\cal D}\) is a set \(T( A , {\cal D} ),\) which is the attractor of an affine iterated function system, satisfying \(T = \cup_{d\in {\cal D}} (T + d).\) It is known that \(T( A , {\cal D} )\) tiles \({\Bbb R}^n\) by some subset of \({\Bbb Z}^n.\)This paper proves that every standard digit set \({\cal D}\) gives a set \(T( A , {\cal D} )\) that tiles \({\Bbb R}^n\) with a lattice tiling.

On recoding in arithmetic algorithms

Recoding (a transformation between digit sets and/or radices) plays a central role in the design of efficient arithmetic algorithms. A suitable recoding reduces both the cost and delay of implementation of arithmetic operations. We present a simplified basis for developing recodings which are useful in digit-recurrence and parallel algorithms for multiplication and division/square-root. >

HANDWRITTEN NUMERAL AND MACHINE PRINTED MULTIPLE FONT CHARACTER RECOGNITION USING NEURAL NETWORK CLASSIFIER

In this paper a new neural network is proposed for recognition of handwritten digits and multi-font machine printed characters. In this system, overlapped regional chain code histograms of characters are used as features and a neural network has been used for classification. A new neural network learning algorithm that combines unsupervised learning with supervised learning has been developed. This new algorithm overcomes the slow learning and difficult convergence problems that are typical of back-propagation learning algorithms. The algorithm was tested on a large set of handwritten digits collected from real world data and a set of multi-font machine printed English letters.

Tapping sensitivity to processing in short-term memory.

This experiment examines the interference between tapping and searching in short-term memory (STM). In a concurrent processing condition, the subject first memorized a set of digits, and then executed a series of 17 taps separated by 2 s subjective time intervals. In each intertap interval, a probe was presented for comparison with the items in the memory set. The amount of processing in STM was varied from series to series by manipulating the number of items in the memory set. The results showed that mean duration and variability of intertap intervals increased proportionally with memory set size. This effect is explained by interference between processing in STM and the timing component of the tapping performance. Tapping sensitivity to memory load suggests that, under some conditions, variations in tapping performance are valid indicators of STM processing requirements of a concurrent primary task.

Integral Self-Affine Tiles in ℝ n I. Standard and Nonstandard Digit Sets

Integral Self-Affine Tiles in ℝ^<i>n</i> I. Standard and Nonstandard Digit Sets

Self-Affine Tiles in [formula omitted

Aself-affine tilein Rnis a setTof positive measure withA(T)=∪d∈D(T+d), whereAis an expandingn×nreal matrix with |det(A)|=man integer, and D={d, d2, ..., dm}⊆Rnis a set ofmdigits. It is known that self-affine tiles always give tilings of Rnby translation. This paper extends known characterizations of digit sets D yielding self-affine tiles. It proves several results about the structure of tilings of Rnpossible using such tiles, and gives examples showing the possible relations between self-replicating tilings and general tilings, which clarify results of Kenyon on self-replicating tilings.

Haar Bases forL2([formula omitted]n) and Algebraic Number Theory

Gröchenig and Madych showed that a Haar-type orthonormal wavelet basis ofL2(Rn) can be constructed from the characteristic functionχQof a setQif and only ifQis an affine image of an integral self-affine tileTwhich tiles Rnusing the integer lattice Zn. An integral self-affine tileT=T(A, D) is the attractor of an iterated function systemT=∪mi=1A−1(T+di) where A∈Mn(Z) is an expandingn×ninteger matrix and the digit set D={d1, d2, …, dm}⊆Znhasm=|det(A)|, provided that the Lebesgue measureμ(T)>0. Two necessary conditions forT(A, D) to tile Rnwith the integer lattice Znare that D be a complete set of coset representatives of Zn/A(Zn) and that Z[A, D]=Zn, where Z[A, D] is the smallest A-invariant lattice containing all {di−dj:i≠j}. These two conditions are necessary and sufficient in the special case that |det(A)|=2. We study these two conditions for an arbitrary matrix A∈Mn(Z). We prove that a digit set D satisfying the two conditions exists whenever |det(A)|⩾n+1. When |det(A)|=2 there are number-theoretic obstructions to the existence of such D. Using these we exhibit a (non-expanding) A∈M2(Z) for which no digit set has Z[A, D]=Z2. However we show that for all expanding integer matrices A in dimensions 2 and 3, there exists some digit set D that satisfies the two conditions. Could this be true for all expanding integer matrices in dimensionsn⩾4? A necessary condition is that the (non-Galois) field[formula]have class number one for alln⩾4.

The Division Algorithm in Complex Bases

AbstractComplex numbers can be represented in positional notation using certain Gaussian integers as bases and digit sets. We describe a long division algorithm to divide one Gaussian integer by another, so that the quotient is a periodic expansion in such a complex base. To divide by the Gaussian integer w in the complex base b, using a digit set D, the remainder must be in the set wT(b,D) ∩ ℤ[i], where T(b,D) is the set of complex numbers with zero integer part in the base. The set T(b,D) tiles the plane, and can be described geometrically as the attractor of an iterated function system of linear maps. It usually has a fractal boundary. The remainder set can be determined algebraically from the cycles in a certain directed graph.

Abnormal lateralization in schizophrenia: empirical evidence for an integrated model

Cerebral deficits in schizophrenia have been associated either with left hemisphere dysfunction, or with impairment in interhemispheric transfer. To test the relative efficacy of the two hypotheses, 63 right-handed subjects (24 paranoid schizophrenics, 21 nonparanoid schizophrenics, and 18 normal controls) were given two auditory lateral tasks: identification of dichotically delivered tonal sequences and digit sets. As the data analyses showed, more correct responses were given by the normal controls than by the patients, and by males than by females. Generally, report (especially from the ear reported second) was better from the right than from the left ear. This right ear advantage (which was more consistent in males than in females) was abnormally large for the paranoid schizophrenics who showed on both tests, relative to normal subjects, a slight decrement in right ear report and a significant decrement in left ear report. The data were interpreted as indicating left hemisphere dysfunction and structural callosal damage in paranoid schizophrenics.