Many studies suggest that EEG signals provide enough information for the detection of human emotions with feature based classification methods. However, very few studies have reported a classification method that reli...Many studies suggest that EEG signals provide enough information for the detection of human emotions with feature based classification methods. However, very few studies have reported a classification method that reliably works for individual participants (classification accuracy well over 90%). Further, a necessary condition for real life applications is a method that allows, irrespective of the immense individual difference among participants, to have minimal variance over the individual classification accuracy. We conducted offline computer aided emotion classification experiments using strict experimental controls. We analyzed EEG data collected from nine participants using validated film clips to induce four different emotional states (amused, disgusted, sad and neutral). The classification rate was evaluated using both unsupervised and supervised learning algorithms (in total seven “state of the art” algorithms were tested). The largest classification accuracy was computed by means of Support Vector Machine. Accuracy rate was on average 97.2%. The experimental protocol effectiveness was further supported by very small variance among individual participants’ classification accuracy (within interval: 96.7%, 98.3%). Classification accuracy evaluated on reduced number of electrodes suggested, consistently with psychological constructionist approaches, that we were able to classify emotions considering cortical activity from areas involved in emotion representation. The experimental protocol therefore appeared to be a key factor to improve the classification outcome by means of data quality improvements.展开更多
Autism spectrum disorder（ASD） encompasses a complex set of developmental neurological disorders,characterized by de？cits in social communication and excessive repetitive behaviors. In recent years, ASD is increasin...Autism spectrum disorder（ASD） encompasses a complex set of developmental neurological disorders,characterized by de？cits in social communication and excessive repetitive behaviors. In recent years, ASD is increasingly being considered as a disease of the synapse.One main type of genetic aberration leading to ASD is gene duplication, and several mouse models have been generated mimicking these mutations. Here, we studied the effects of MECP2 duplication and human chromosome15q11-13 duplication on synaptic development and neural circuit wiring in the mouse sensory cortices. We showed that mice carrying MECP2 duplication had speci？c defects in spine pruning, while the 15q11-13 duplication mouse model had impaired spine formation. Our results demonstrate that spine pathology varies signi？cantly between autism models and that distinct aspects of neural circuit development may be targeted in different ASD mutations.Our results further underscore the importance of gene dosage in normal development and function of the brain.展开更多
We emphasize the importance of studying the primate brain in cognitive neuroscience and suggest a new mind-set in primate experimentation within the boundaries of animal welfare regulations.Specifically,we list the ad...We emphasize the importance of studying the primate brain in cognitive neuroscience and suggest a new mind-set in primate experimentation within the boundaries of animal welfare regulations.Specifically,we list the advantages of investigating both genes and neural mechanisms and processes in the emergence of behavioral and cognitive functions,and propose the establishment of an open field of primate research.The latter may be conducted by implementing and harmonizing experimental practices with ethical guidelines that regulate(1)management of natural parks with free-moving populations of target nonhuman primates,(2)establishment of indoor-outdoor labs for both system genetics and neuroscience investigations,and(3)hotel space and technologies which remotely collect and dislocate information regarding primates geographically located elsewhere.展开更多
Objective: Development of an EEG preprocessing technique for improvement of de tection of Alzheimer’s disease (AD).The technique is based on filtering of EEG data using blind source separation (BSS) and projection of...Objective: Development of an EEG preprocessing technique for improvement of de tection of Alzheimer’s disease (AD).The technique is based on filtering of EEG data using blind source separation (BSS) and projection of components which are possibly sensitive to cortical neuronal impairment found in early stages of AD. Methods: Artifact-free 20 s intervals of raw resting EEG recordings from 22 pat ients with Mild Cognitive Impairment (MCI) who later proceeded to AD and 38 age -matched normal controls were decomposed into spatio-temporally decorrelated c omponents using BSS algorithm‘AMUSE’. Filtered EEG was obtained by back projec tion of components with the highest linear predictability. Relative power of fil tered data in delta, theta, alpha 1, alpha 2, beta1, and beta 2 bands were proce ssed with Linear Discriminant Analysis (LDA). Results: Preprocessing improved th e percentage of correctly classified patients and controls computed with jack-k nifing cross-validation from 59 to 73%and from 76 to 84%, correspondingly. Co nclusions: The proposed approach can significantly improve the sensitivity and s pecificity of EEG based diagnosis. Significance: Filtering based on BSS can impr ove the performance of the existing EEG approaches to early diagnosis of Alzheim er’s disease. It may also have potential for improvement of EEG classification in other clinical areas or fundamental research. The developed method is quite g eneral and flexible, allowing for various extensions and improvements.展开更多
The preoptic area （POA） is located in the most anterior part of the hypothalamus and is bordered dorsally by the anterior commissure and anteroventrally by the nucleus of the diagonal band of Broca[1]. Accumulating ...The preoptic area （POA） is located in the most anterior part of the hypothalamus and is bordered dorsally by the anterior commissure and anteroventrally by the nucleus of the diagonal band of Broca[1]. Accumulating evidence from developmental neurobiology suggests, however, that the POA may be a separate entity from hypothalamus, and may actually be part of the basal telencephalon[2,3]. Both the hypothalamus and POA are highly complex and heterogeneous areas, containing multiple nuclei, each of which has specific fundamental functions for survival. Among these, the POA contains nuclei involved in the regulation of blood osmolality and temperature （the median preoptic nucleus）, sleep （the ventrolateral preoptic and suprachiasmatic nuclei）, ovulation （gonadotropin-releasing hormone neurons scattered mainly in the ventral part of the POA）, male sexual behavior （the medial preoptic nucleus）, and parental behavior （the central part of the medial POA, cMPOA）.展开更多
The cerebellum expresses one of the highest levels of the plasma membrane Ca2+ATPase,isoform 2 in the mammalian brain.This highly efficient plasma membrane calcium transporter protein is enriched within the main outpu...The cerebellum expresses one of the highest levels of the plasma membrane Ca2+ATPase,isoform 2 in the mammalian brain.This highly efficient plasma membrane calcium transporter protein is enriched within the main output neurons of the cerebellar cortex;i.e. the Purkinje neurons(PNs) .Here we review recent evidence,including electrophysiological and calcium imaging approaches using the plasma membrane calcium ATPase 2(PMCA2) knockout mouse,to show that PMCA2 is critical for the physiological control of calcium at cerebellar synapses and cerebellar dependent behaviour.These studies have also revealed that deletionof PMCA2 throughout cerebellar development in the PMCA2 knockout mouse leads to permanent signalling and morphological alterations in the PN dendrites. Whilst these findings highlight the importance of PMCA2 during cerebellar synapse function and development,they also reveal some limitations in the use of the PMCA2 knockout mouse and the need for additional experimental approaches including cell-specific and reversible manipulation of PMCAs.展开更多
SLC6A4 （solute carrier family 6, member 4） gene encodes a serotonin transporter （5-hydroxytryptamine transporter, HTT）, which transports synaptic serotonin into presynaptic terminal. SLC6A4 is known to be the targ...SLC6A4 （solute carrier family 6, member 4） gene encodes a serotonin transporter （5-hydroxytryptamine transporter, HTT）, which transports synaptic serotonin into presynaptic terminal. SLC6A4 is known to be the target of antidepressants such as selective serotonin reuptake inhibitors （SSRIs）. Inhibition of HTT increases synaptic serotonin concentration and thereby exerts antidepressant efficacy. A large number of genetic studies suggest the contribution of genetic variations of SLC6A4 to various psychiatric disorders. The most studied genetic variation, HTT-linked polymorphic region （HTTLPR）,展开更多
We propose a generalized Lanczos method to generate the many-body basis states of quantum lattice models using tensor-network states （TNS）. The ground-state wave function is represented as a linear superposition com...We propose a generalized Lanczos method to generate the many-body basis states of quantum lattice models using tensor-network states （TNS）. The ground-state wave function is represented as a linear superposition composed from a set of TNS generated by Lanczos iteration. This method improves significantly the accuracy of the tensor-network algorithm and provides an effective way to enlarge the maximal bond dimension of TNS. The ground state such obtained contains significantly more entanglement than each individual TNS, reproducing correctly the logarithmic size dependence of the entanglement entropy in a critical system. The method can be generalized to non-Hamiltonian systems and to the calculation of low-lying excited states, dynamical correlation functions, and other physical properties of strongly correlated systems.展开更多
Purpose: This study aimed to examine whether the signal-to-noise ratio (SNR) could be increased by combining integrated slice-by-slice shimming (iShim) with a fat suppression (FS) method other than short-tau inversion...Purpose: This study aimed to examine whether the signal-to-noise ratio (SNR) could be increased by combining integrated slice-by-slice shimming (iShim) with a fat suppression (FS) method other than short-tau inversion recovery (STIR) in diffusion-weighted imaging (DWI) and q-space imaging (qsi). Methods: We acquired DWI images (b-values: 0 and nine steps from 400 to 10,000 s/mm2 for six axes) using a prototypical single-shot echo planar imaging sequence by combining two types of shimming (3D Shim and iShim) and two types of FS (STIR and water excitation [WE]) in 10 volunteers. In the DWI study, the SNR for each b-value, FS effect in the b0 image, and distortion in the added image (b0 - b10,000) were evaluated for the above-mentioned four imaging methods. qsi involved original DWI images. In the qsi study, the SNR was evaluated. Results: With regard to both 3D Shim and iShim, the SNRs were significantly higher when using WE than when using STIR in b0 - b900 images (p Conclusion: The combination of iShim and WE has a high SNR on qsi.展开更多
Brain computer interface (BCI) aims at creating new communication channels without depending on brain’s normal output channels of peripheral nerves and muscles. However,natural and sophisticated interactions manner b...Brain computer interface (BCI) aims at creating new communication channels without depending on brain’s normal output channels of peripheral nerves and muscles. However,natural and sophisticated interactions manner between brain and computer still remain challenging. In this paper,we investigate how the duration of event-related desynchronization/synchronization (ERD/ERS) caused by motor im-agery (MI) can be modulated and used as an additional control parameter beyond simple binary deci-sions. Furthermore,using the non-time-locked properties of sustained (de)synchronization,we have developed an asynchronous BCI system for driving a car in 3D virtual reality environment (VRE) based on cumulative incremental control strategy. The extensive real time experiments confirmed that our new approach is able to drive smoothly a virtual car within challenging VRE only by the MI tasks with-out involving any muscular activities.展开更多
Autism spectrum disorder(ASD) is gathering concerns in socially developed countries. ASD is a neuropsychiatric disorder of genetic origin with high prevalence of 1%–2%. The patients with ASD characteristically show i...Autism spectrum disorder(ASD) is gathering concerns in socially developed countries. ASD is a neuropsychiatric disorder of genetic origin with high prevalence of 1%–2%. The patients with ASD characteristically show impaired social skills. Today, many genetic studies identify numerous susceptible genes and genetic loci associated with ASD. Although some genetic factors can lead to abnormal brain function linked to ASD phenotypes, the pathogenic mechanism of ASD is still unclear. Here, we discuss a new mouse model for ASD as an advanced tool to understand the mechanism of ASD.展开更多
This paper presents a simple technique to fabricate new electrofluidic devices for the three-dimensional(3D)manipulation of microorganisms by hybrid subtractive and additive femtosecond(fs)laser microfabrication(fs la...This paper presents a simple technique to fabricate new electrofluidic devices for the three-dimensional(3D)manipulation of microorganisms by hybrid subtractive and additive femtosecond(fs)laser microfabrication(fs laser-assisted wet etching of glass followed by water-assisted fs laser modification combined with electroless metal plating).The technique enables the formation of patterned metal electrodes in arbitrary regions in closed glass microfluidic channels,which can spatially and temporally control the direction of electric fields in 3D microfluidic environments.The fabricated electrofluidic devices were applied to nanoaquariums to demonstrate the 3D electro-orientation of Euglena gracilis(an elongated unicellular microorganism)in microfluidics with high controllability and reliability.In particular,swimming Euglena cells can be oriented along the z-direction(perpendicular to the device surface)using electrodes with square outlines formed at the top and bottom of the channel,which is quite useful for observing the motions of cells parallel to their swimming directions.Specifically,z-directional electric field control ensured efficient observation of manipulated cells on the front side(45 cells were captured in a minute in an imaging area of~160×120μm),resulting in a reduction of the average time required to capture the images of five Euglena cells swimming continuously along the z-direction by a factor of~43 compared with the case of no electric field.In addition,the combination of the electrofluidic devices and dynamic imaging enabled observation of the flagella of Euglena cells,revealing that the swimming direction of each Euglena cell under the electric field application was determined by the initial body angle.展开更多
The present article gives an introduction to information geometry and surveys its applications in the area of machine learning,optimization and statistical inference.Information geometry is explained intuitively by us...The present article gives an introduction to information geometry and surveys its applications in the area of machine learning,optimization and statistical inference.Information geometry is explained intuitively by using divergence functions introduced in a manifold of probability distributions and other general manifolds.They give a Riemannian structure together with a pair of dual flatness criteria.Many manifolds are dually flat.When a manifold is dually flat,a generalized Pythagorean theorem and related projection theorem are introduced.They provide useful means for various approximation and optimization problems.We apply them to alternative minimization problems,Ying-Yang machines and belief propagation algorithm in machine learning.展开更多
文摘Many studies suggest that EEG signals provide enough information for the detection of human emotions with feature based classification methods. However, very few studies have reported a classification method that reliably works for individual participants (classification accuracy well over 90%). Further, a necessary condition for real life applications is a method that allows, irrespective of the immense individual difference among participants, to have minimal variance over the individual classification accuracy. We conducted offline computer aided emotion classification experiments using strict experimental controls. We analyzed EEG data collected from nine participants using validated film clips to induce four different emotional states (amused, disgusted, sad and neutral). The classification rate was evaluated using both unsupervised and supervised learning algorithms (in total seven “state of the art” algorithms were tested). The largest classification accuracy was computed by means of Support Vector Machine. Accuracy rate was on average 97.2%. The experimental protocol effectiveness was further supported by very small variance among individual participants’ classification accuracy (within interval: 96.7%, 98.3%). Classification accuracy evaluated on reduced number of electrodes suggested, consistently with psychological constructionist approaches, that we were able to classify emotions considering cortical activity from areas involved in emotion representation. The experimental protocol therefore appeared to be a key factor to improve the classification outcome by means of data quality improvements.
基金supported by grants from the Strategic Priority Research Program of the Chinese Academy of Sciences,China(XDB02010000)the National Natural Science Foundation of China(31530030 and 81371270)the Science and Technology Commission of Shanghai Municipality,China(16XD1404800)
文摘Autism spectrum disorder（ASD） encompasses a complex set of developmental neurological disorders,characterized by de？cits in social communication and excessive repetitive behaviors. In recent years, ASD is increasingly being considered as a disease of the synapse.One main type of genetic aberration leading to ASD is gene duplication, and several mouse models have been generated mimicking these mutations. Here, we studied the effects of MECP2 duplication and human chromosome15q11-13 duplication on synaptic development and neural circuit wiring in the mouse sensory cortices. We showed that mice carrying MECP2 duplication had speci？c defects in spine pruning, while the 15q11-13 duplication mouse model had impaired spine formation. Our results demonstrate that spine pathology varies signi？cantly between autism models and that distinct aspects of neural circuit development may be targeted in different ASD mutations.Our results further underscore the importance of gene dosage in normal development and function of the brain.
文摘We emphasize the importance of studying the primate brain in cognitive neuroscience and suggest a new mind-set in primate experimentation within the boundaries of animal welfare regulations.Specifically,we list the advantages of investigating both genes and neural mechanisms and processes in the emergence of behavioral and cognitive functions,and propose the establishment of an open field of primate research.The latter may be conducted by implementing and harmonizing experimental practices with ethical guidelines that regulate(1)management of natural parks with free-moving populations of target nonhuman primates,(2)establishment of indoor-outdoor labs for both system genetics and neuroscience investigations,and(3)hotel space and technologies which remotely collect and dislocate information regarding primates geographically located elsewhere.
文摘Objective: Development of an EEG preprocessing technique for improvement of de tection of Alzheimer’s disease (AD).The technique is based on filtering of EEG data using blind source separation (BSS) and projection of components which are possibly sensitive to cortical neuronal impairment found in early stages of AD. Methods: Artifact-free 20 s intervals of raw resting EEG recordings from 22 pat ients with Mild Cognitive Impairment (MCI) who later proceeded to AD and 38 age -matched normal controls were decomposed into spatio-temporally decorrelated c omponents using BSS algorithm‘AMUSE’. Filtered EEG was obtained by back projec tion of components with the highest linear predictability. Relative power of fil tered data in delta, theta, alpha 1, alpha 2, beta1, and beta 2 bands were proce ssed with Linear Discriminant Analysis (LDA). Results: Preprocessing improved th e percentage of correctly classified patients and controls computed with jack-k nifing cross-validation from 59 to 73%and from 76 to 84%, correspondingly. Co nclusions: The proposed approach can significantly improve the sensitivity and s pecificity of EEG based diagnosis. Significance: Filtering based on BSS can impr ove the performance of the existing EEG approaches to early diagnosis of Alzheim er’s disease. It may also have potential for improvement of EEG classification in other clinical areas or fundamental research. The developed method is quite g eneral and flexible, allowing for various extensions and improvements.
文摘The preoptic area （POA） is located in the most anterior part of the hypothalamus and is bordered dorsally by the anterior commissure and anteroventrally by the nucleus of the diagonal band of Broca[1]. Accumulating evidence from developmental neurobiology suggests, however, that the POA may be a separate entity from hypothalamus, and may actually be part of the basal telencephalon[2,3]. Both the hypothalamus and POA are highly complex and heterogeneous areas, containing multiple nuclei, each of which has specific fundamental functions for survival. Among these, the POA contains nuclei involved in the regulation of blood osmolality and temperature （the median preoptic nucleus）, sleep （the ventrolateral preoptic and suprachiasmatic nuclei）, ovulation （gonadotropin-releasing hormone neurons scattered mainly in the ventral part of the POA）, male sexual behavior （the medial preoptic nucleus）, and parental behavior （the central part of the medial POA, cMPOA）.
基金Supported by An Health Research Council-Japanese Society for the Promotion of Science fellowship and the Neurological Foundation of New Zealand(Empson RM and Nagaraja RY)a British Biological and Biotechnology Research Council award(Empson RM and Garside ML)+1 种基金a Department of Physiology MSc studentship(Huang H)RIKEN intramural funding(Knpfel T)
文摘The cerebellum expresses one of the highest levels of the plasma membrane Ca2+ATPase,isoform 2 in the mammalian brain.This highly efficient plasma membrane calcium transporter protein is enriched within the main output neurons of the cerebellar cortex;i.e. the Purkinje neurons(PNs) .Here we review recent evidence,including electrophysiological and calcium imaging approaches using the plasma membrane calcium ATPase 2(PMCA2) knockout mouse,to show that PMCA2 is critical for the physiological control of calcium at cerebellar synapses and cerebellar dependent behaviour.These studies have also revealed that deletionof PMCA2 throughout cerebellar development in the PMCA2 knockout mouse leads to permanent signalling and morphological alterations in the PN dendrites. Whilst these findings highlight the importance of PMCA2 during cerebellar synapse function and development,they also reveal some limitations in the use of the PMCA2 knockout mouse and the need for additional experimental approaches including cell-specific and reversible manipulation of PMCAs.
文摘SLC6A4 （solute carrier family 6, member 4） gene encodes a serotonin transporter （5-hydroxytryptamine transporter, HTT）, which transports synaptic serotonin into presynaptic terminal. SLC6A4 is known to be the target of antidepressants such as selective serotonin reuptake inhibitors （SSRIs）. Inhibition of HTT increases synaptic serotonin concentration and thereby exerts antidepressant efficacy. A large number of genetic studies suggest the contribution of genetic variations of SLC6A4 to various psychiatric disorders. The most studied genetic variation, HTT-linked polymorphic region （HTTLPR）,
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11190024 and 11474331)
文摘We propose a generalized Lanczos method to generate the many-body basis states of quantum lattice models using tensor-network states （TNS）. The ground-state wave function is represented as a linear superposition composed from a set of TNS generated by Lanczos iteration. This method improves significantly the accuracy of the tensor-network algorithm and provides an effective way to enlarge the maximal bond dimension of TNS. The ground state such obtained contains significantly more entanglement than each individual TNS, reproducing correctly the logarithmic size dependence of the entanglement entropy in a critical system. The method can be generalized to non-Hamiltonian systems and to the calculation of low-lying excited states, dynamical correlation functions, and other physical properties of strongly correlated systems.
文摘Purpose: This study aimed to examine whether the signal-to-noise ratio (SNR) could be increased by combining integrated slice-by-slice shimming (iShim) with a fat suppression (FS) method other than short-tau inversion recovery (STIR) in diffusion-weighted imaging (DWI) and q-space imaging (qsi). Methods: We acquired DWI images (b-values: 0 and nine steps from 400 to 10,000 s/mm2 for six axes) using a prototypical single-shot echo planar imaging sequence by combining two types of shimming (3D Shim and iShim) and two types of FS (STIR and water excitation [WE]) in 10 volunteers. In the DWI study, the SNR for each b-value, FS effect in the b0 image, and distortion in the added image (b0 - b10,000) were evaluated for the above-mentioned four imaging methods. qsi involved original DWI images. In the qsi study, the SNR was evaluated. Results: With regard to both 3D Shim and iShim, the SNRs were significantly higher when using WE than when using STIR in b0 - b900 images (p Conclusion: The combination of iShim and WE has a high SNR on qsi.
基金Supported by the National High-Tech Research Program of China (Grant No.2006AA01Z125)the National Basic Research Program of China (Grant No.2005CB724301)
文摘Brain computer interface (BCI) aims at creating new communication channels without depending on brain’s normal output channels of peripheral nerves and muscles. However,natural and sophisticated interactions manner between brain and computer still remain challenging. In this paper,we investigate how the duration of event-related desynchronization/synchronization (ERD/ERS) caused by motor im-agery (MI) can be modulated and used as an additional control parameter beyond simple binary deci-sions. Furthermore,using the non-time-locked properties of sustained (de)synchronization,we have developed an asynchronous BCI system for driving a car in 3D virtual reality environment (VRE) based on cumulative incremental control strategy. The extensive real time experiments confirmed that our new approach is able to drive smoothly a virtual car within challenging VRE only by the MI tasks with-out involving any muscular activities.
基金supported in part by a grant from Japan Society for the Promotion of Science and Ministry of Education,Culture,Sports,Science and Technology KAKENHI,Core Research for Evolutional Science and Technology,Strategic International Cooperative Program of Japan Science and Technology Agency,and by research grants from Takeda Science Foundation,the Naito Foundation and Terumo Life Science Foundation
文摘Autism spectrum disorder(ASD) is gathering concerns in socially developed countries. ASD is a neuropsychiatric disorder of genetic origin with high prevalence of 1%–2%. The patients with ASD characteristically show impaired social skills. Today, many genetic studies identify numerous susceptible genes and genetic loci associated with ASD. Although some genetic factors can lead to abnormal brain function linked to ASD phenotypes, the pathogenic mechanism of ASD is still unclear. Here, we discuss a new mouse model for ASD as an advanced tool to understand the mechanism of ASD.
文摘This paper presents a simple technique to fabricate new electrofluidic devices for the three-dimensional(3D)manipulation of microorganisms by hybrid subtractive and additive femtosecond(fs)laser microfabrication(fs laser-assisted wet etching of glass followed by water-assisted fs laser modification combined with electroless metal plating).The technique enables the formation of patterned metal electrodes in arbitrary regions in closed glass microfluidic channels,which can spatially and temporally control the direction of electric fields in 3D microfluidic environments.The fabricated electrofluidic devices were applied to nanoaquariums to demonstrate the 3D electro-orientation of Euglena gracilis(an elongated unicellular microorganism)in microfluidics with high controllability and reliability.In particular,swimming Euglena cells can be oriented along the z-direction(perpendicular to the device surface)using electrodes with square outlines formed at the top and bottom of the channel,which is quite useful for observing the motions of cells parallel to their swimming directions.Specifically,z-directional electric field control ensured efficient observation of manipulated cells on the front side(45 cells were captured in a minute in an imaging area of~160×120μm),resulting in a reduction of the average time required to capture the images of five Euglena cells swimming continuously along the z-direction by a factor of~43 compared with the case of no electric field.In addition,the combination of the electrofluidic devices and dynamic imaging enabled observation of the flagella of Euglena cells,revealing that the swimming direction of each Euglena cell under the electric field application was determined by the initial body angle.
文摘The present article gives an introduction to information geometry and surveys its applications in the area of machine learning,optimization and statistical inference.Information geometry is explained intuitively by using divergence functions introduced in a manifold of probability distributions and other general manifolds.They give a Riemannian structure together with a pair of dual flatness criteria.Many manifolds are dually flat.When a manifold is dually flat,a generalized Pythagorean theorem and related projection theorem are introduced.They provide useful means for various approximation and optimization problems.We apply them to alternative minimization problems,Ying-Yang machines and belief propagation algorithm in machine learning.
