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Regularization on discrete spaces
 Pattern Recognition
, 2005
"... Abstract. We consider the classification problem on a finite set of objects. Some of them are labeled, and the task is to predict the labels of the remaining unlabeled ones. Such an estimation problem is generally referred to as transductive inference. It is wellknown that many meaningful inductive ..."
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Cited by 41 (1 self)
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Abstract. We consider the classification problem on a finite set of objects. Some of them are labeled, and the task is to predict the labels of the remaining unlabeled ones. Such an estimation problem is generally referred to as transductive inference. It is wellknown that many meaningful inductive or supervised methods can be derived from a regularization framework, which minimizes a loss function plus a regularization term. In the same spirit, we propose a general discrete regularization framework defined on finite object sets, which can be thought of as discrete analogue of classical regularization theory. A family of transductive inference schemes is then systemically derived from the framework, including our earlier algorithm for transductive inference, with which we obtained encouraging results on many practical classification problems. The discrete regularization framework is built on discrete analysis and geometry developed by ourselves, in which a number of discrete differential operators of various orders are constructed, which can be thought of as discrete analogues of their counterparts in the continuous case. 1
TRACE THEOREMS FOR TREES AND APPLICATION TO THE HUMAN LUNGS
"... (Communicated by Axel Klar) Abstract. The aim of this paper is to develop a model of the respiratory system. The real bronchial tree is embedded within the parenchyma, and ventilation is caused by negative pressures at the alveolar level. We aim to describe the series of pressures at alveolae in the ..."
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(Communicated by Axel Klar) Abstract. The aim of this paper is to develop a model of the respiratory system. The real bronchial tree is embedded within the parenchyma, and ventilation is caused by negative pressures at the alveolar level. We aim to describe the series of pressures at alveolae in the form of a function, and to establish a sound mathematical framework for the instantaneous ventilation process. To that end, we treat the bronchial tree as an infinite resistive tree, we endow the space of pressures at bifurcating nodes with the natural energy norm (rate of dissipated energy), and we characterise the pressure field at its boundary (i.e. set of simple paths to infinity). In a second step, we embed the infinite collection of leafs in a bounded domain Ω ⊂ Rd, and we establish some regularity properties for the corresponding pressure field. In particular, for the infinite counterpart of a regular, healthy lung, we show that the pressure field lies in a Sobolev space Hs(Ω), with s ≈ 0.45. This allows us to propose a model for the ventilation process that takes the form of a boundary problem,
THE FIRST L pCOHOMOLOGY OF SOME FINITELY GENERATED GROUPS AND pHARMONIC FUNCTIONS
, 2005
"... Abstract. Let G be a finitely generated infinite group and let p> 1. In this paper we make a connection between the first L pcohomology space of G and pharmonic functions on G. We also describe the elements in the first L pcohomology space of groups with polynomial growth, and we give an inclu ..."
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Abstract. Let G be a finitely generated infinite group and let p> 1. In this paper we make a connection between the first L pcohomology space of G and pharmonic functions on G. We also describe the elements in the first L pcohomology space of groups with polynomial growth, and we give an inclusion result for nonamenable groups. 1.
MAXIMUM PRINCIPLE AND COMPARISON PRINCIPLE OF pHARMONIC FUNCTIONS VIA pHARMONIC BOUNDARY OF GRAPHS
"... Abstract. We prove the maximum principle and the comparison principle of pharmonic functions via pharmonic boundary of graphs. By applying the comparison principle, we also prove the solvability of the boundary value problem of pharmonic functions via pharmonic boundary of graphs. 1. ..."
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Abstract. We prove the maximum principle and the comparison principle of pharmonic functions via pharmonic boundary of graphs. By applying the comparison principle, we also prove the solvability of the boundary value problem of pharmonic functions via pharmonic boundary of graphs. 1.