Information Fusion and Hierarchical Knowledge Discovery by ARTMAP Neural Networks

Mapping novel terrain from sparse, complex data often requires the resolution of conflicting information from sensors working at different times, locations, and scales, and from experts with different goals and situations. Information fusion methods help resolve inconsistencies in order to distinguish correct from incorrect answers, as when evidence variously suggests that an object's class is car, truck, or airplane. The methods developed here consider a complementary problem, supposing that information from sensors and experts is reliable though inconsistent, as when evidence suggests that an objects class is car, vehicle, or man-made. Underlying relationships among objects are assumed to be unknown to the automated system of the human user. The ARTMAP information fusion system uses distributed code representations that exploit the neural network's capacity for one-to-many learning in order to produce self-organizing expert systems that discover hierarchial knowledge structures. The system infers multi-level relationships among groups of output classes, without any supervised labeling of these relationships. The procedure is illustrated with two image examples.Air Force Office of Scientific Research (F49620-01-1-0397, F49620-01-1-0423); Office of Naval Research (N00014-01-1-0624

Samuel Holland: From Gunner and Sapper to Surveyor-General 1755-1764

The British Army engaged, in 1755, the young Dutch officer, Samuel Holland (whose patron was already the Third Duke of Richmond), to serve in North America as an artillery and engineering subaltern. Following many months’ service directly under the field commander, Holland became deeply involved in the siege of Louisbourg (1758) as the engineering assistant to James Wolfe. The latter warmly commended Holland to Richmond for his superior efficiency and his bravery under constantly heavy enemy fire. After the siege, Holland drew an accurate plan of the fortified port, illustrating the steps of the siege-attack and defence. He became busy in 1758 and 1759 in the preparation of the British attack on Quebec, during which he met the famous British navigator, James Cook, with whom he exchanged expertise. At the siege of Quebec he continued to serve Wolfe until the latter’s death in the battle of September 1759. From then until 1762 Holland served James Murray, first as part of a team of engineers participating in the defence of Quebec against a French siege, during which he was named acting chief engineer in place of a wounded officer and eventually confined in the city with the rest of the garrison until the siege was raised by the Royal Navy. Thereafter, under Murray’s command, Holland’s main achievement was his part in the surveying and mapping of the St. Lawrence valley, leading to the production of the “Murray Map”, an immense contribution to eighteenth-century cartography. Murray vehemently held, in the face of claims by officers of the Royal Engineers, that Samuel Holland deserved the most credit for the success and high quality of the product. During the Seven Years War, Holland had been promoted Captain. Excluded from the Royal Engineers, he was therefor quite independent of the bureaucracy of that corps when in 1763 he sought-in new American colonies ceded by France-an appointment in surveying and cartography. As a guest in the London house of the Duke of Richmond he had the opportunity of meeting influential politicians, where the recognition by Wolfe and Murray of the high quality of his professional competence finally led the British government to appoint him Surveyor General in North America

On spherical averages of radial basis functions

A radial basis function (RBF) has the general form $$s(x)=\sum_{k=1}^{n}a_{k}\phi(x-b_{k}),\quad x\in\mathbb{R}^{d},$$ where the coefficients a 1,…,a n are real numbers, the points, or centres, b 1,…,b n lie in ℝ d , and φ:ℝ d →ℝ is a radially symmetric function. Such approximants are highly useful and enjoy rich theoretical properties; see, for instance (Buhmann, Radial Basis Functions: Theory and Implementations, [2003]; Fasshauer, Meshfree Approximation Methods with Matlab, [2007]; Light and Cheney, A Course in Approximation Theory, [2000]; or Wendland, Scattered Data Approximation, [2004]). The important special case of polyharmonic splines results when φ is the fundamental solution of the iterated Laplacian operator, and this class includes the Euclidean norm φ(x)=‖x‖ when d is an odd positive integer, the thin plate spline φ(x)=‖x‖2log  ‖x‖ when d is an even positive integer, and univariate splines. Now B-splines generate a compactly supported basis for univariate spline spaces, but an analyticity argument implies that a nontrivial polyharmonic spline generated by (1.1) cannot be compactly supported when d>1. However, a pioneering paper of Jackson (Constr. Approx. 4:243–264, [1988]) established that the spherical average of a radial basis function generated by the Euclidean norm can be compactly supported when the centres and coefficients satisfy certain moment conditions; Jackson then used this compactly supported spherical average to construct approximate identities, with which he was then able to derive some of the earliest uniform convergence results for a class of radial basis functions. Our work extends this earlier analysis, but our technique is entirely novel, and applies to all polyharmonic splines. Furthermore, we observe that the technique provides yet another way to generate compactly supported, radially symmetric, positive definite functions. Specifically, we find that the spherical averaging operator commutes with the Fourier transform operator, and we are then able to identify Fourier transforms of compactly supported functions using the Paley–Wiener theorem. Furthermore, the use of Haar measure on compact Lie groups would not have occurred without frequent exposure to Iserles’s study of geometric integration

Why are we here

John 1:29-4