Marie Eliasen

Dr Janne Schurmann Tolstrup

Anne-Marie Nybo Andersen

M Gronbaek

Jorn Olsen

Katrine Strandberg-Larsen

Anne-Marie Nybo Andersen

Morten Grønbaek

Jørn Olsen

Prenatal alcohol exposure and autistic spectrum disorder

Keywords: Pregnancy, Autistic Spectrum Disorder, Alcohol, Children 'Pregnancy', 'Alcohol Drinking', 'Child Development Disorders, Pervasive' Pregnancy, alcohol, autistic spectrum disorders, children

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers,

 [START_REF] Who | ICD-10: International Statistical Classification of Diseases and Related Health Problems, 10th Revision[END_REF]The prevalence of ASD has been increasing in the last two decades, [START_REF] Lauritsen | The incidence and prevalence of pervasive developmental disorders: a Danish population-based study[END_REF] but the reason for this increase is unknown and the aetiology of ASD is still mostly unknown. A strong genetic component has been established, but also the age of the parents, psychiatric disorders, obstetrical complications, and some teratogenic factors such as thalidomide have been associated with ASD. [START_REF] Arndt | The teratology of autism[END_REF][START_REF] Bello | Autism and environmental influences: review and commentary[END_REF][START_REF] Casanova | The neuropathology of autism[END_REF][START_REF] Hultman | Perinatal risk factors for infantile autism[END_REF][START_REF] Kolevzon | Prenatal and perinatal risk factors for autism: a review and integration of findings[END_REF][START_REF] Newschaffer | The epidemiology of autism spectrum disorders[END_REF] Two case studies and a small follow-up study have reported higher frequency of ASD among children with foetal alcohol effects, [START_REF] Aronson | Attention deficits and autistic spectrum problems in children exposed to alcohol during gestation: a follow-up study[END_REF][START_REF] Harris | Autistic behaviors in offspring of mothers abusing alcohol and other drugs: a series of case reports[END_REF][START_REF] Nanson | Autism in fetal alcohol syndrome: a report of six cases[END_REF] but a causal relation between prenatal alcohol exposure and ASD is not established. The brains of children with ASD have some similarities as do the brains of animals prenatally exposed to alcohol e.g. a decreased number of purkinje cells and reduced neuronal size compared to the normal brain. [START_REF] Casanova | The neuropathology of autism[END_REF][START_REF] Ikonomidou | Ethanol-induced apoptotic neurodegeneration and fetal alcohol syndrome[END_REF][START_REF] Pierce | Purkinje cell vulnerability to developmental ethanol exposure in the rat cerebellum[END_REF][START_REF] West | Cell population depletion associated with fetal alcohol brain damage: mechanisms of BAC-dependent cell loss[END_REF] Furthermore, a high prenatal alcohol exposure is associated with impaired neurodevelopment in humans. [START_REF] Jacobson | Effects of prenatal alcohol exposure on child development[END_REF][START_REF] Sayal | Prenatal alcohol exposure and gender differences in childhood mental health problems: A longitudinal population-based study[END_REF][START_REF] Streissguth | Neurobehavioral effects of prenatal alcohol: Part III. PLS analyses of neuropsychologic tests[END_REF][START_REF] Streissguth | Moderate prenatal alcohol exposure: effects on child IQ and learning problems at age 7 1/2 years[END_REF] Hence, it is possible that prenatal alcohol exposure increases the risk of ASD. The central nervous system develops throughout the entire pregnancy, and specific functions are developed at specific time points. Prenatal alcohol exposure may therefore have time dependent effects on the foetal brain. Furthermore, the detrimental effects of alcohol also seem to depend on the achieved blood alcohol level, [START_REF] Ikonomidou | Ethanol-induced apoptotic neurodegeneration and fetal alcohol syndrome[END_REF][START_REF] Pierce | Purkinje cell vulnerability to developmental ethanol exposure in the rat cerebellum[END_REF][START_REF] Streissguth | Neurobehavioral effects of prenatal alcohol: Part III. PLS analyses of neuropsychologic tests[END_REF][START_REF] Jacobson | Relation of maternal age and pattern of pregnancy drinking to functionally significant cognitive deficit in infancy[END_REF][START_REF] Pierce | Blood alcohol concentration: a critical factor for producing fetal alcohol effects[END_REF][START_REF] Henderson | Systematic review of the fetal effects of prenatal bingedrinking[END_REF][START_REF] Kesmodel | Binge drinking in pregnancy--frequency and methodology[END_REF][START_REF] Maier | Fetal alcohol exposure and temporal vulnerability: regional differences in cell loss as a function of the timing of binge-like alcohol exposure during brain development[END_REF][START_REF] Maier | Regional differences in cell loss associated with binge-like alcohol exposure during the first two trimesters equivalent in the rat[END_REF][START_REF] Streissguth | Prenatal alcohol and offspring development: the first fourteen years[END_REF] but evidence from human studies is sparse. Previously, we have found that exposure to binge drinking (intake of larger doses of alcohol at a single occasion) in gestational week 11-16 is associated with an increased risk of neonatal seizures and epilepsy in childhood. [START_REF] Sun | Binge drinking during pregnancy and risk of seizures in childhood: a study based on the Danish National Birth Cohort[END_REF] Binge drinking is a regular part of the alcohol drinking pattern among younger people in many societies, and in Norway and Denmark it has been shown that 25 of pregnant women, respectively, have binge drunk once during pregnancy. [START_REF] Kesmodel | Use of alcohol and illicit drugs among pregnant Danish women, 1998[END_REF][START_REF] Strandberg-Larsen | Characteristics of women who binge drink before and after they become aware of their pregnancy[END_REF] Moreover, binge drinking is increasing in the United States among both pregnant and non-pregnant women. [START_REF] Ethen | Alcohol Consumption by Women Before and During Pregnancy[END_REF] In this study we examined whether weekly alcohol consumption, number of binge drinking episodes, as well as binge drinking during different developmental time periods is associated with the incidence of autistic spectrum disorders or infantile autism in the offspring.

Methods

Study population

The Danish National Birth Cohort (DNBC) is a population-based cohort of pregnant women and their offspring approved by the Danish Scientific Ethical Committees. [START_REF] Olsen | The Danish National Birth Cohort--its background, structure and aim[END_REF] The women were recruited from March 1996 to November 2002 through their first antenatal visit with their general practitioner. Around 50 % of all practitioners in Denmark participated and around 60 % of those invited returned a signed consent form and thereby consented to be telephone-interviewed twice during pregnancy, around pregnancy week 17 (quartile 14-20) and pregnancy week 32 (quartile 30-34). English translations of the interviews are available at www.dnbc.dk. Of all enrolled pregnancies (n=100,418), we limited our analyses to women who gave birth to a live-born singleton and completed the first interview (n=86,783). We excluded women with incomplete information on binge drinking (n=330), women with missing information on gestational age at the first interview (n=878) as well as women for whom the discrepancy between the selfreported gestational age and the information recorded in the National Discharge Registry exceeded one week (n=4441) as one of the main exposures in this study was gestational timing of binge drinking. Furthermore, we excluded women with lacking information on the covariates included in the regression models (n=580) and children who were unidentifiable in the National Central Personal Register (n=2) probably due to a coding error of their personal id number. Hereby 80,552 children and their mothers were eligible for the analyses.

Alcohol consumption

In the first interview, the women were asked to report the number of drinks of beer, wine, and spirits per week they on average consumed while being pregnant. One drink was defined as one bottle of beer (0.33L), one glass of wine, or one glass of spirits, and in Denmark the alcohol content Women who reported to consume less than one drink per week of these beverages were assigned the numeric value of ½ drink per week. The weekly intake of beer, wine, and spirits were added up to a total and before doing any analyses categorized into the groups: 0, ½-1½, 2-3½ and 4 or more drinks per week. In analyses with too few cases, the two last groups were combined into one group: 2 or more drinks per week.

Information on binge drinking was obtained by asking the women how many times and when in pregnancy they had consumed five or more alcoholic drinks on a single occasion since the onset of pregnancy, defined as the first day of the last menstrual period. Information on binge drinking during the period from onset of pregnancy to the first interview was given in the first interview, while information from the second interview, if available, was used to describe binge drinking in the period between the two interviews. We categorized number of binge episodes into 0, 1, 2 and 3 or more episodes. For the analyses of the timing of binge drinking we classified the pregnancy into different fetal developmental time periods and categorized binge drinking as 'yes' versus 'no' in the preconceptional period (pregnancy weeks 0-2), the fertilization and implementation period (weeks 3-4), the embryonic period (weeks 5-10), the early fetal period (weeks 11-16), and the mid fetal period (weeks 17-30), respectively. The classification was set before doing the analyses and was defined as detailed as the number of exposed permitted.

Identification of ASD and infantile autism

Using the civil registry number, which is unique to every Danish citizen, diagnoses of ASD or infantile autism on the children were identified through linkage to the Danish Psychiatric Central Register. [START_REF] Munk-Jorgensen | The Danish Psychiatric Central Register[END_REF] The registry contains information about all contacts to psychiatric inpatient and outpatient facilities since 1995 and contacts are diagnosed according to the International [START_REF] Lauritsen | The incidence and prevalence of pervasive developmental disorders: a Danish population-based study[END_REF] The children were classified as having ASD if they were diagnosed with one of the following ICD-10 codes: Infantile autism (F84.0), atypical autism (F84.1), Asperger's syndrome (F84.5), or pervasive developmental disorder not otherwise specified (F84.8 and F84.9), and classified as having infantile autism if they were diagnosed with the ICD-10 code F84.0. The children classified as having infantile autism were therefore a subset of those classified as having ASD.

Covariates

Based on the existing literature, we made an a priori decision to adjust for maternal age (<25, 25-29, 30-34, 35-39, ≥40 years), paternal age (father unidentified, <25, 25-29, 30-34, 35-39, 40-44, ≥45 years), smoking habits during pregnancy (non-smoker, stopped smoking before time of the interview, 1-10, 11 or more cigarettes per day), self-reported maternal history of psychiatric disorder (affective disorders, nervous conditions, psychosis), household occupational position (higher-grade professionals, middle-grade professionals, skilled workers, unskilled workers, students, unemployed for more than one year), and parity (0, 1, 2 or more children). [START_REF] Kesmodel | Use of alcohol and illicit drugs among pregnant Danish women, 1998[END_REF][START_REF] Ethen | Alcohol Consumption by Women Before and During Pregnancy[END_REF][START_REF] Larsson | Risk factors for autism: perinatal factors, parental psychiatric history, and socioeconomic status[END_REF][START_REF] Croen | Descriptive epidemiology of autism in a California population: who is at risk?[END_REF][START_REF] Fombonne | Epidemiological surveys of autism and other pervasive developmental disorders: an update[END_REF][START_REF] Harrison | Alcohol and Drug Use Before and During Pregnancy: An Examination of Use Patterns and Predictors of Cessation[END_REF] Information on the covariates was obtained from the first interview.

Statistics

Data were analysed by means of Cox regression models with the children's age in days as underlying time variable, using SAS, 9.1 (SAS Institute Inc, Cary, NC). The children were followed from the date of birth until the date of diagnosis with ASD, death, emigration, or the end of follow up (February 10, 2008), whichever came first. Robust standard errors were calculated to account for the dependency between children with siblings in the study (n=4943). The number of weeks with information on binge drinking was included in the strata-statement in all analyses on binge drinking

F o r R e v i e w O n l y 8
to account for the fact that the women were interviewed in different pregnancy weeks and subsequent had information on binge drinking from different spans of pregnancy. The proportional hazard assumption was examined for average alcohol consumption and binge drinking by plotting the negative logarithm of the estimated survival function against survival time (age) and by including an interaction term between age and average alcohol consumption and binge drinking, respectively, in the models. No sign of violation of the proportional hazards assumption was seen.

Initially, we estimated crude and adjusted hazard ratios for ASD and infantile autism according to average alcohol consumption and the number of binge episodes during pregnancy in separate models as well as in one combined model. Furthermore, the hazard ratios for ASD according to the timing of binge episodes were estimated by including the time periods for binge drinking simultaneously into one model. The association between the timing of binge drinking and infantile autism was not investigated due to the lack of statistical power. Eight percent of the binge drinkers did not report the timing of at least one of their binge episodes. We imputed the missing information on timing by use of a probability distribution of when in pregnancy binge drinking occurred based on the existing data on timing of binge drinking. For example, if a woman was interviewed in pregnancy week 14 and reported two binge episodes, with the first episode in pregnancy week 4 and the timing of the second was missing, the second episode was defined to be between week 4 and 14. The probability distribution was made dependent on whether the pregnancy was recognized before pregnancy week 4, 4-8, 8-12, after week 12, or unknown. The imputation procedure was repeated five times. The presented estimates are the average of the estimates, and the used standard errors are based on the variation both within and between the imputations. [START_REF] Barnard | Multiple imputation methods[END_REF] To examine the robustness of the obtained results, the analyses were restricted to primiparous women, who conceived within six months of trying (n=19,253). Hereby we attempted to avoid bias arising from selective change in behaviours related to the women's previous reproductive experience or prolonged waiting time to pregnancy. 37 Furthermore, we restricted the analyses to women who had complete information on the time of binge episodes (n=78,881) and to the first enrolled pregnancies (n=75,609). The restrictions were only made on the analyses for ASD due to the limited number of cases of infantile autism. Almost half of the women reported an average weekly intake of at least half a standard alcoholic drink (45%), and approximately one fourth of the women reported at least one episode of binge drinking during pregnancy (table 1). Binge drinking most often occurred before the pregnancy was recognized. Only 5.4 % of all the women reported binge drinking subsequent to pregnancy recognition. Other characteristics of the women are described in table 1.

(Table 1 here) Women who reported an average alcohol consumption of ½-1½, 2-3½, or 4 or more drinks per week during pregnancy had a risk of having a child diagnosed with ASD or infantile autism similar to what was found for women who reported no average alcohol consumption during pregnancy (table 2). Women who reported binge drinking once during pregnancy had an adjusted hazard ratio (HR) for having a child with ASD of 0.72 (95 % confidence interval (CI): 0.53 to 0.97) compared to women who reported no episodes of binge drinking during pregnancy. One episode of binge drinking was also associated with a lower hazard of infantile autism when compared to no binge drinking (HR: 0.61; 95% CI: 0.36 to 1.02). For two or more binge episodes the hazard ratios for ASD and infantile autism were around 1 compared with no binge drinking. Adjustment for parental age, smoking habits, self-reported maternal history of psychiatric disorder, household sociooccupational position, and parity let only to minor changes of the estimates (table 2).

(Table 2 Women with one or more binge episodes in pregnancy weeks 0-2, 3-4, 5-10, and 17-30 had basically a similar hazard of having a child with ASD as women who did not binge drink in the same period (table 3). However, binge drinking in pregnancy weeks 11-16 yield a hazard ratio above 1 compared to non-binge drinking, but the confidence interval still included 1 (adjusted HR: 1.32; 95% CI: 0.65 to 2.68). Adjustment for potential covariates, average alcohol consumption and the number of binge episodes did not change the estimates much.

(Table 3 here)

The lower hazard ratio for ASD associated with one binge episode increased to 0.84 (95 % CI: 0.51 to 1.36) in the analyses restricted to primiparous women who conceived within six months of trying (table 4). Restriction to women with complete information on the time of binge episodes and restriction to the first enrolled pregnancies did not change the estimates essentially.

(Table 4

Discussion

In this large population-based cohort study we found no positive associations between ASD and infantile autism and average alcohol consumption, number, or timing of binge episodes during pregnancy. Binge drinking once during pregnancy was associated with a lower risk of having a child with ASD which could be a chance finding or be confounded. Women with a high risk of reproductive failure may modify their drinking behaviour in subsequent pregnancies. If reproductive failures or prolonged waiting time to pregnancy are associated with increased risk of ASD, and if the study does not fully adjust for the underlying causes of these reproductive failures or prolonged waiting time, confounding will be present. The increasing of the hazard ratio for one binge episode when restricting the analysis to primiparous women who conceived within six months of trying may indicate that confounding is present in the analyses of the total population.

We have not adjusted for ethnicity, which has been shown to be associated with both ASD and alcohol consumption. [START_REF] Hultman | Perinatal risk factors for infantile autism[END_REF][START_REF] Ethen | Alcohol Consumption by Women Before and During Pregnancy[END_REF][START_REF] Lauritsen | Effects of familial risk factors and place of birth on the risk of autism: a nationwide register-based study[END_REF] However, availability to take part in interviews hold in Danish was a condition for participating in the study, hence we expected only few non-Danish born women not well integrated in the Danish society to be included in the study. Consequently, we assume that our findings are not likely to be confounded by ethnicity.

The analyses on the timing of binge drinking may indicate a possible susceptible period for binge drinking in weeks 11-16. However, the confidence interval included 1 and it may very well be a chance finding. The results were based on only ten cases which limits the statistical power. Hence, further investigations on the potential effects of the timing of binge drinking are needed to make further conclusions.

The present study is to our knowledge the first to examine the association between moderate to light prenatal alcohol exposure and ASD. A suggested higher prevalence of ASD among children with fetal alcohol effects and fetal alcohol syndrome found in two case studies and a small follow- with a heavy drinking pattern that could be of importance. [START_REF] Aronson | Attention deficits and autistic spectrum problems in children exposed to alcohol during gestation: a follow-up study[END_REF][START_REF] Harris | Autistic behaviors in offspring of mothers abusing alcohol and other drugs: a series of case reports[END_REF][START_REF] Nanson | Autism in fetal alcohol syndrome: a report of six cases[END_REF] Our study does not rule out that heavy alcohol intake may cause ASD or that alcohol exposure in short time windows may play a role as we do not have the data to rule out such effects.

The detrimental effects of alcohol may depend on the maximal blood alcohol concentration that is achieved during a drinking occasion. [START_REF] Ikonomidou | Ethanol-induced apoptotic neurodegeneration and fetal alcohol syndrome[END_REF][START_REF] Pierce | Purkinje cell vulnerability to developmental ethanol exposure in the rat cerebellum[END_REF][START_REF] Streissguth | Neurobehavioral effects of prenatal alcohol: Part III. PLS analyses of neuropsychologic tests[END_REF][START_REF] Jacobson | Relation of maternal age and pattern of pregnancy drinking to functionally significant cognitive deficit in infancy[END_REF][START_REF] Pierce | Blood alcohol concentration: a critical factor for producing fetal alcohol effects[END_REF][START_REF] Henderson | Systematic review of the fetal effects of prenatal bingedrinking[END_REF][START_REF] Kesmodel | Binge drinking in pregnancy--frequency and methodology[END_REF][START_REF] Maier | Fetal alcohol exposure and temporal vulnerability: regional differences in cell loss as a function of the timing of binge-like alcohol exposure during brain development[END_REF][START_REF] Maier | Regional differences in cell loss associated with binge-like alcohol exposure during the first two trimesters equivalent in the rat[END_REF][START_REF] Streissguth | Prenatal alcohol and offspring development: the first fourteen years[END_REF] If so, our results would be conservative estimates of this association since binge drinking is not an accurate measure of the maximal blood alcohol concentration because the amount of time that is spend during drinking differs. However, information on the pace of drinking was not given and we could not address this point any further.

Our study has several strengths including its size and linkage to nationwide registries ensuring almost complete follow-up. Other strengths include the measurement of alcohol consumption during pregnancy with validated questionnaires, both including beverage-specific average alcohol consumption and repeated measures of the number and timing of binge drinking episodes. [START_REF] Kesmodel | Binge drinking in pregnancy--frequency and methodology[END_REF][START_REF] Kesmodel | Self reported alcohol intake in pregnancy: comparison between four methods[END_REF][START_REF] Strandberg-Larsen | Do women give the same information on binge drinking during pregnancy when asked repeatedly?[END_REF] Moreover, the setting of the study in a culture where alcohol consumption among women is more socially accepted than in many other countries makes the social desirability of underreporting less likely, and thereby probably makes the data more reliable. However, we do expect pregnant women to underreport their alcohol consumption to some extent. [START_REF] Kesmodel | Self reported alcohol intake in pregnancy: comparison between four methods[END_REF] This would probably bias the estimates towards the null value, as we do not expect the misclassification of alcohol consumption to be associated with the risk of having a child with ASD since the alcohol intake was collected in pregnancy.

We expect that some children with ASD and infantile autism were not diagnosed even though the validity of infantile autism in the Danish Psychiatric Central Register is presumed to be high. [START_REF] Madsen | MMR vaccination and autism--a population-based follow-up study[END_REF] Some of the children fulfilling the criteria for ASD may not have been diagnosed because of less 14 severe behavioural problems that do not lead to contact with the psychiatric hospitals, which are the only places in Denmark where ASD is diagnosed, except for one private diagnostic service which does not report to the Danish Psychiatric Central Register. [START_REF] Lauritsen | The incidence and prevalence of pervasive developmental disorders: a Danish population-based study[END_REF] The conservative estimate of the incidence of ASD may have biased the results, but the direction of the bias is unknown. However, mothers with extensive alcohol consumption might be less likely to seek psychiatric help for their children as extensive alcohol consumption has been associate with reduced ability to assess the mental development of the child. [START_REF] Seagull | Maternal assessment of infant development: associations with alcohol and drug use in pregnancy[END_REF] A differential misclassification might therefore be present which could mask a positive association between alcohol consumption and ASD. However, we expect that only few women with extensive alcohol consumption participated in the study. Furthermore, the women with a moderate alcohol consumption in our study had a higher than average education and socio-occupational position. [START_REF] Strandberg-Larsen | Characteristics of women who binge drink before and after they become aware of their pregnancy[END_REF] The possible misclassification is therefore expected to be of minor importance.

Only around 30 % of all pregnant women in Denmark during the study period participated in the DNBC (60 % of those invited), and thereby selection bias is possible. [START_REF] Olsen | The Danish National Birth Cohort--its background, structure and aim[END_REF] Pilot tests have shown that the main reasons for non-participation were lack of time and interest. [START_REF] Olsen | The Danish National Birth Cohort--its background, structure and aim[END_REF] It is likely that alcohol consumption during pregnancy influenced the participation, but as the risk of ASD was not known beforehand, we assume that participation was not associated with ASD. Hence, it is believed that our results are not highly influenced by selection bias. However, selection bias could occur if women with a high alcohol intake who suffered from ASD were less likely to accept the invitation, since the offspring of these mothers would have a higher probability of having ASD. Such a selection could mask an association when family history of ASD was not adjusted for. In conclusion, we did not find that average alcohol consumption, frequency or timing of binge drinking episodes during pregnancy were associated with the risk of having a child with ASD or infantile autism for women drinking light to moderate. ASD is a rare condition and despite this large cohort study, the results were limited by statistical power.

  disorders (ASD) including infantile autism are pervasive developmental disorders characterized by qualitative impairments in social interaction and communication and stereotyped repetitive behaviour.

  of drink is roughly the same and equivalent to approximately 12 g of pure alcohol.

  , Tenth Edition (ICD-10).

  80,552 pregnant women, 401 gave birth to a child diagnosed with ASD and 157 to a child diagnosed with infantile autism. The average age of the children at the time of diagnosis was 4.9 years for ASD and 4.1 years for infantile autism. At the end of follow up the children were between 4.6 and 10.3 years, with an average age of 7.4 years.

  be explained by excessive prenatal exposure to alcohol or by other factors associated

  Katrine Strandberg-Larsen is the guarantor for the paper. Katrine Strandberg-Larsen and Marie Eliasen have checked the references for accuracy and completeness.

Table 1

 1 Maternal and paternal characteristics according to alcohol consumption during pregnancy

		Total N (%)	Maternal age (Years) Median (10 th ; 90 th percentile)	Paternal age (Years) Median (10 th ; 90 th percentile)	Maternal smokers a (%)	(%) Maternal psychiatric disorder b	(%) Middle or professionals higher grade	Primiparous (%)
	Total population Total population Total population Total population	80,552 (100)	30 (25; 36)	32 (27;39)	25.5	7.7	55.2	46.8
	Average alcohol consumption Average alcohol consumption Average alcohol consumption Average alcohol consumption
	Drinks per week
	0	44,493 (55.2)	29 (24; 35)	32 (26; 40)	26.0	8.1	51.1	49.4
	½-1½	26,663 (33.1)	30 (25; 36)	33 (27; 39)	23.3	7.1	59.3	45.3
	2-3½	7,708 (9.6)	31 (26; 37)	34 (28; 41)	27.6	7.2	63.9	40.6
	4+	1,688 (2.1)	33 (27; 38)	35 (29; 43)	38.3	10.7	60.5	33.0
	Binge Binge Binge Binge drinking drinking drinking drinking
	Number of episodes
	0	59,609 (74.0)	30 (25; 36)	33 (27; 39)	22.3	7.5	55.2	43.4
	1	13,037 (16.2)	29 (25; 35)	32 (26; 39)	30.2	7.7	55.6	54.3
	2	4,699 (5.8)	29 (24; 35)	32 (26; 39)	37.8	8.2	53.8	60.4
	3+	3,207 (4.0)	30 (24; 36)	32 (27; 40)	47.6	12.0	55.4	61.6
	Timing of binge episodes Timing of binge episodes Timing of binge episodes Timing of binge episodes	b b b b

Table 2

 2 Crude and adjusted hazard ratios for ASD and infantile autism according to alcohol consumption during pregnancy. † Crude ‡ Adjusted for maternal age, paternal age, smoking habits, self-reported maternal history of psychiatric disorder, household sociooccupational position, and parity

			ASD ASD ASD ASD			Infantil Infantil Infantil Infantile autism e autism e autism e autism
	Exposure Exposure Exposure Exposure	Cases	HR † (95 % CI) HR ‡ (95 % CI)	Cases (%) HR † (95 % CI) HR ‡ (95 % CI)
	Average alcohol consumption Average alcohol consumption Average alcohol consumption Average alcohol consumption
	Drinks per week
	0	237	1 (ref.)	1 (ref.)	92 (0.2)	1 (ref.)	1 (ref.)
	½-1½	119	0.83 (0.67-1.04)	0.84 (0.68-1.05)	45 (0.2)	0.81 (0.57-1.16)	0.79 (0.56-1.13)
	2-3½	36	0.87 (0.62-1.24)	0.86 (0.60-1.23)	17 (0.2)	1.06 (0.63-1.77)	0.98 (0.58-1.65)
	4 +	9	0.97 (0.50-1.90)	0.91 (0.46-1.80)	3 (0.2)	0.84 (0.27-2.66)	0.71 (0.22-2.30)
	Binge drinking Binge drinking Binge drinking Binge drinking
	Episodes
	0	303	1 (ref.)	1 (ref.)	125 (0.2)	1 (ref.)	1 (ref.)
	1	51	0.74 (0.55-1.00)	0.72 (0.53-0.97)	17 (0.1)	0.61 (0.37-1.02)	0.61 (0.36-1.02)
	2	31	1.33 (0.84-1.77)	1.16 (0.80-1.68)	7 (0.2)	0.69 (0.32-1.49)	0.68 (0.32-1.46)
	3+	16	0.91 (0.55-1.50)	0.80 (0.48-1.34)	8 (0.3)	1.16 (0.57-2.37)	1.04 (0.48-2.22)

Table 3

 3 Crude and adjusted hazard ratios for ASD according to timing of binge drinking episodes. Adjusted for maternal age, paternal age, smoking habits, self-reported maternal history of psychiatric disorder, household sociooccupational position, and parity a Missing indicates women who had no information on binge drinking in the weeks investigated due to last interview before the specific week

			ASD ASD ASD ASD
	Timing of episodes Timing of episodes Timing of episodes Timing of episodes	Cases	HR † (95 % CI)	HR ‡ (95 % CI)
	Weeks 0 Weeks 0 Weeks 0 Weeks 0----2 2 2 2
	0	370	1 (ref.)	1 (ref.)
	1+	31	0.94 (0.65-1.36)	0.90 (0.62-1.31)
	Weeks 3 Weeks 3 Weeks 3 Weeks 3----4 4 4 4
	0	380	1 (ref.)	1 (ref.)
	1+	51	0,83 (0.61-1.12)	0.79 (0.58-1.07)
	Weeks 5 Weeks 5 Weeks 5 Weeks 5----10 10 10 10
	0	372	1 (ref.)	1 (ref.)
	1+	29	1.02 (0.67-1.56)	0.99 (0.65-1.52)
	Missing a	0	-	-
	Weeks 11 Weeks 11 Weeks 11 Weeks 11----16 16 16 16
	0	379	1 (ref.)	1 (ref.)
	1+	10	1.40 (0.71-2.78)	1.32 (0.65-2.68)
	Missing a	12	-	-
	Weeks 17 Weeks 17 Weeks 17 Weeks 17----30 30 30 30
	0	249	1 (ref.)	1 (ref.)
	1+	8	1.05 (0.52-2.15)	1.02 (0.49-2.11)
	Missing a	144	-	-
	† Crude
	‡

Table 4

 4 Adjusted hazard ratios for ASD and infantile autism according to alcohol consumption during pregnancy for first-time pregnant women who waited less than six months to become pregnant (N=19,253) † Crude ‡ Adjusted for maternal age, paternal age, smoking habits, self-reported maternal history of psychiatric disorder, household sociooccupational position, and parity

			ASD ASD ASD ASD
		Cases	HR † (95 % CI)	HR ‡ (95 % CI)
	Average alcohol consumption Average alcohol consumption Average alcohol consumption Average alcohol consumption
	Drinks per week
	0	62	1 (ref.)	1 (ref.)
	½-1½	28	0.82 (0.52-1.28)	0.85 (0.55-1.34)
	2+	15	1.46 (0.83-2.57)	1.49 (0.84-2.66)
	Binge drinking Binge drinking Binge drinking Binge drinking
	Episodes
	0	68	1 (ref.)	1 (ref.)
	1	20	0.91 (0.55-1.49)	0.92 (0.56-1.53)
	2	10	1.11 (0.58-2.15)	1.08 (0.57-2.04)
	3+	7	1.08 (0.50-2.36)	1.00 (0.44-2.28)

Acknowledgements

This work was supported by grants from the Danish Ministry of Health, the Danish National Board of Health, the Health Insurance Foundation, and the National Institute of Public Health, University of Southern Denmark. The Danish National Research Foundation has established the Danish Epidemiology Science Centre that initiated and created the Danish National Birth Cohort. The cohort is furthermore a result of major grant from this foundation. Additional support for the Danish National Birth Cohort was obtained from the Pharmacy Foundation, the Egmont Foundation, the March of Dimes Birth Defects Foundation, and the Augustinus Foundation.

Conflicts of interests

All authors declare that they have no conflicts of interest and no financial interest of the manuscript.

Key messages

• The average alcohol consumption, frequency or timing of binge drinking episodes during pregnancy were not found to be associated with the risk of having a child with ASD or infantile autism for women drinking light to moderate.

• Further studies are needed to rule out any effect of prenatal alcohol exposure on ASD and infantile autism especially for women drinking heavier and for high blood alcohol concentration in short time windows.

• When studying the effects of prenatal alcohol exposure it is important to include binge drinking and the timing of the binge drinking episodes.


J J Verbeek
email: jverbeek@science.uva.nl

N Vlassis

intelligent autonomous systems

Keywords: Gaussian fields, regression, correspondence learning, active learning, model selection

Gaussian fields (GF) have recently received considerable attention for dimension reduction and semi-supervised classification. This paper presents two contributions. First, we show how the GF framework can be used for regression tasks on high-dimensional data. We consider an active learning strategy based on entropy minimization and a maximum likelihood model selection method. Second, we show how a recent generalization of the Locally Linear Embedding algorithm for correspondence learning can also be cast into the GF framework, which obviates the need to choose a representation dimensionality.

Introduction

Regression on the basis of high-dimensional data is a challenging problem in machine learning. In order to successfully learn regression functions either a limited class of regression models must be used (e.g. only linear models) or large quantities of data are required to reliably find a good model from a larger class. Recently several techniques have been proposed for unsupervised dimension reduction and semi-supervised classification that are based on a nearest neighbor graph on the set of high-dimensional 'inputs'. The graph on the data is used to specify the assumption that nearest neighbors (according to some similarity measure) in the high-dimensional input space will have similar 'outputs'. In case of dimension reduction [START_REF] Roweis | Nonlinear dimensionality reduction by locally linear embedding[END_REF][START_REF] Tenenbaum | A global geometric framework for nonlinear dimensionality reduction[END_REF][START_REF] Niyogi | Laplacian eigenmaps for dimensionality reduction and data representation[END_REF] the outputs are the desired lowdimensional representations of the inputs, in the case of semi-supervised classification [START_REF] Joachims | Transductive learning via spectral graph partitioning[END_REF][START_REF] Zhu | Combining active learning and semi-supervised learning using gaussian fields and harmonic functions[END_REF][START_REF] Belkin | Semi-supervised learning on riemannian manifolds[END_REF] the outputs are class labels.

These techniques are based on a non-negative energy function that sums terms that are quadratic in the outputs of neighbors in the graph. With the energy function we can associate a Gaussian field (GF); a probability density function on all outputs. The GF density associated with a quadratic energy function is proportional to the exponent of the negative energy function (therefore it follows that the density is a Gaussian). The methods mentioned above return a vector of outputs that minimizes the energy function; this is the mean of the corresponding GF density.

In this paper we consider GFs defined on nearest neighbor graphs for two different learning problems. The first problem is semi-supervised regression on high-dimensional data that exhibits a low-dimensional structure. For example, the input data can be the pixel values of images of a face that looks in different directions and we want to map these images to pose parameters. If we consider the raw images as input vectors then the input equals the number of pixels in the images, but the data are confined to an embedding of a low-dimensional manifold that contains the images of the face looking in different directions. In this setting, unsupervised data (for which we do not know the pose parameters) can be exploited to identify the lowdimensional manifold structure of the data. Due to the low-dimensionality of the manifold, only few supervised examples are needed to successfully learn a regression function to predict the pose parameters. The amount of supervised data that can be acquired in practice is often limited. This motivates active learning approaches, which determine on the basis of a set of unsupervised data (and possibly some supervised data), which unsupervised data are expected to yield the most information if the supervised signal would be available for them.

The second problem is correspondence learning. In this problem two high-dimensional data sets are given that share the same underlying modes of variability. For example, each set can be a collection of images of different views of a particular object. The manifolds on which the two data sets lie, parameterized by the shared modes of variability, can be aligned if some 'correspondences' are given. A correspondence is a pair that consists of a point in the first data set and one in the second, which are known to share the same underlying low-dimensional coordinate (e.g. images of the first and second object viewed from the same direction, see Fig. 4). The goal is now to predict for a data point in one set the corresponding observation in the other set. This problem is similar to regression, except that in this case the 'output' is also high dimensional. Also in this setting, the low-dimensional manifolds can be recovered from unsupervised data, which can then be aligned using only relatively few correspondences. Moreover, as opposed to earlier work on correspondence learning [START_REF] Ham | Learning high dimensional correspondences from low dimensional manifolds[END_REF][START_REF] Verbeek | Non-linear CCA and PCA by Alignment of Local Models[END_REF], in the GF framework we do not need to specify a dimensionality of the underlying manifold. This is advantageous since the dimensionality is not always known for the problem at hand.

The main difference between the GF approach and more traditional approaches lies in how we specify a density over the outputs. For GFs, this density model is determined by the nearest neighbor graph on the inputs; in this manner the density is derived from -and reflects the density of-the given inputs. More traditional techniques also define a density on the outputs, although often implicitly. They do this by limiting attention to a parametric class of functions (e.g. linear functions or feed-forward neural networks) that map inputs to outputs. The choice for a specific class of functions is often motivated by reasons of computational efficiency rather than by the suitability of this class for the given problem.

In the next section we introduce GFs defined on nearest neighbor graphs. In section Section 3 we focus on regression. First, we consider a maximum likelihood method to select an appropriate number of nearest neighbors to specify the neighborhood graph. Second, we consider an entropy minimization based query selection procedure for active learning. We present experimental results that show the effectiveness of both procedures. In Section 4 we turn to correspondence learning using GFs. We propose a new approach for the correspondence learning based on GFs, and compare it to an existing approach based on the LLE algorithm. We emphasize an attractive property of GFs for correspondence learning, namely that correspondences can be predicted without an explicit low-dimensional representation as required by the LLE based approach. We conclude with a discussion in Section 5.

Gaussian fields for unsupervised and semi-supervised learning

In this section we briefly summarize how Gaussian Fields have been used before for dimension reduction and semi-supervised classification. Given a set of high-dimensional 'inputs' X = {x i } n i=1 , the first step in defining the GF is to find the nearest neighbors of each x i . Throughout this we define nearest neighbors on the basis of Euclidean distance. We note that it is not necessary to naively compute all pairwise squared distances to find the nearest neighbors, see e.g. [START_REF] Karger | Finding nearest neighbors in growth-restricted metrics[END_REF].

Let y i denote the scalar 'output' of x i , and let y denote the n-dimensional vector with all outputs. Given the nearest neighbors, we define an energy function E(y) over the outputs by summing over all input pairs x i and x j which are nearest neighbors, the squared difference of their outputs (y i -y j ) 2 times a weighting factor a ij . If we let a ij = 0 only for nearest neighbors, then:

E(y) = n i=1 n j=1 a ij (y i -y j ) 2 = y (D -A)y = y Ly, (1)
where the degree matrix D is a diagonal matrix containing the row sums of the adjacency matrix A. The matrix L is known as the graph Laplacian of A. Note that if ∀ i,j a ij ≥ 0 then ∀ y E(y) ≥ 0, however the converse is not true: matrices A with negative entries exist with ∀ y E(y) ≥ 0. For example, the Locally Linear Embedding (LLE) algorithm [START_REF] Roweis | Nonlinear dimensionality reduction by locally linear embedding[END_REF] uses

A = W + W -W W, (2)
where each row of W sums to 1 and w ij = 0 only for nearest neighbors. Using this adjacency matrix D = I and the energy E(y) can be written as:

E(y) = n i=1 (y i - n j=1 w ij y j ) 2 = (I -W)y 2 F ≥ 0, (3)
where • F denotes the Frobenius norm of a matrix (i.e. the square root of the sum of the square of all elements). The LLE algorithm and the Laplacian Eigenmap algorithm [START_REF] Niyogi | Laplacian eigenmaps for dimensionality reduction and data representation[END_REF] (the latter uses an adjacency matrix with non-negative entries) perform dimension reduction by solving for y that minimizes E(y). Note that there is a trivial solution y = 1c, which is an eigenvector of L with eigenvalue zero. In order to prevent the trivial solution we can minimize the energy under the constraint that y is zero mean and has unit variance. The minimizer is then given by the eigenvector of L with the second smallest eigenvalue. To extract (d + 1) rather than d features, one can minimize E under the constraint that y d+1 should be orthogonal to the already extracted eigenvectors y i (i = 1, . . . , d). The resulting minimizing vectors y i are all eigenvectors of L, the eigenvalues corresponding to later added features exceeding those corresponding to the already extracted features.

For semi-supervised classification, see e.g. [START_REF] Joachims | Transductive learning via spectral graph partitioning[END_REF][START_REF] Zhu | Combining active learning and semi-supervised learning using gaussian fields and harmonic functions[END_REF][START_REF] Belkin | Semi-supervised learning on riemannian manifolds[END_REF], the value of some outputs y i (i.e. the class labels, being 0 or 1 in the two class problems) of some data points x i are known and thus fixed to these values. The energy E(y) is then minimized with respect to the remaining entries of y. Without loss of generality we can assume that all indices of supervised points are smaller than those of the unsupervised points. We can thus use the partition y = [y s y u ] , where y u denotes the sub-vector for unsupervised points and y s the sub-vector for the supervised points. Let L ss , L us and L uu denote the corresponding blocks of L:

L = L ss L su L us L uu . (4)
Throughout we will use this subscripting to refer to blocks of matrices and vectors that correspond to supervised and unsupervised points. The energy can be expanded as:

E(y) = y u L uu y u + y s L ss y s + 2y u L us y s , (5)
and for fixed y s , the minimizer y * u of E(y) with respect to y u , is characterized by the linear equation:

L uu y * u = -L us y s . (6)
Although L uu is sparse, its inverse may be full and for large n computing and even storing the inverse can be problematic. Some authors [START_REF] Joachims | Transductive learning via spectral graph partitioning[END_REF][START_REF] Belkin | Semi-supervised learning on riemannian manifolds[END_REF] constrain y to be in the span of the m eigenvectors of L with smallest eigenvalues, i.e. y = Vw where V contains the m eigenvectors as columns. The coefficients in w are set to minimize the squared error y s -V s w 2 , where V s contains the rows of V corresponding to the supervised points. We note here that an alternative (and exact) method to solve [START_REF] Karger | Finding nearest neighbors in growth-restricted metrics[END_REF], that avoids explicit inversion and that is in practice often much faster than computing a small eigenbasis, is to use the Cholesky decomposition L uu = R R, then solve for R z = -L us y s , and then for Ry * u = z. Several quantities which need to be evaluated in the techniques developed in the next sections can be computed using Cholesky factors in a similar manner.

In practice, the energy function E(y) cannot be directly used to specify a GF since L is singular (the constant vector is an eigenvector with eigenvalue zero). By adding a regularizer we ensure that it is positive definite and can be used as an inverse covariance matrix of the GF. Let M = L + αI, then it is easy to see that M has the same eigenvectors as L, with the eigenvalues increased by α. The regularizer replaces the infinite variance in the direction 1 from the GF distribution with variance α -1 .

For the given points X we define the GF density over outputs y as:

p(y) = N (y; 0, β -1 C) ∝ exp(- β 2 y My), (7)
where β is a scale parameter that controls the smoothness of the density and C denotes the inverse of M. Note that in the GF formulation, fixing some of the outputs to y s induces a conditional density p(y u |y s ). The mean y * u of this conditional density is given by (6) if we replace the blocks of L with the corresponding blocks of M. Thus the methods considered above can be interpreted as computing the mean of the corresponding Gaussian field conditioned on the supervised points. In the next section we will exploit several other quantities associated with the GF density.

Active semi-supervised regression with Gaussian fields

Previous work [START_REF] Roweis | Nonlinear dimensionality reduction by locally linear embedding[END_REF][START_REF] Niyogi | Laplacian eigenmaps for dimensionality reduction and data representation[END_REF][START_REF] Joachims | Transductive learning via spectral graph partitioning[END_REF][START_REF] Zhu | Combining active learning and semi-supervised learning using gaussian fields and harmonic functions[END_REF][START_REF] Belkin | Semi-supervised learning on riemannian manifolds[END_REF] has considered the use of GFs for semi-supervised classification and dimension reduction; here we consider GFs for semi-supervised regression. The GF density p(y) defined in the previous section can be regarded as a prior density over outputs that favors the outputs of neighboring inputs to be similar. Given the values of some of the variables, collected in y s , we obtain the posterior on the remaining variables p(y u |y s ) (in the previous section we already discussed how to compute the mean of the posterior). It is straightforward to generalize this setup to predict several response variables independently using the same Gaussian field; for clarity we focus here on the case with a single response variable.

In work by others on semi-supervised classification, the adjacency matrix A that determines the energy function was specified directly by putting non-negative entries for nearest neighbors, or by using Gaussian kernels. As we argue next, for regression it is more useful to define A through a weight matrix W, as in [START_REF] Golub | Matrix computations[END_REF]. We set w ij = 1/k if x j is among the k nearest neighbors of x i and w ij = 0 otherwise. In this way the energy function sums for each point the squared difference of its output and the average of its neighbors, cf. [START_REF] Ham | Learning high dimensional correspondences from low dimensional manifolds[END_REF]. In comparison, if we directly set the entries A corresponding to nearest neighbors to 1/k then the energy function sums for each point the average squared difference between its output and that of its neighbors.

The apparently small difference to define A has an important consequence for extrapolation performance of regression using the GF. The effect is best understood through a simple example illustrated in Fig. 1. The input data consists of 300 data points along a spiral. The desired output of each point is its position if the data points are numbered along the spiral. For three data points (shown as circles) the desired output is specified. When A is specified directly, there is no force that drives extrapolation: if all points toward the end of the spiral are assigned the same output as the last supervised point, the contribution to the energy induced by these points is zero. When A is defined in the LLE-based manner (2) extrapolation is obtained: the data points towards the end of the spiral are assigned higher values to ensure that the average at the last supervised point matches the supervised value as good as possible. This is necessary since the unsupervised points at the other side of the last supervised point are assigned smaller values for the interpolation between the supervised points.

Because of the lack of extrapolation in the direct method, we use the LLE based energy function [START_REF] Ham | Learning high dimensional correspondences from low dimensional manifolds[END_REF] in the remainder of this paper, with w ij = 1/k when x j is one of the k neighbors of x i . Note that the extrapolation effect does not appear in the case of semi-supervised classification, since then desired outputs are always in the range of the supervised outputs (i.e. between zero and one).

Model selection

The GF density ( 7) is parameterized by k, the number of nearest neighbors connected to each point, and the variance scale parameter β. Since it is hard to set these parameters by hand, it is desirable to find optimal values automatically. We can use the maximum a-posteriori (MAP) criterion to select appropriate values for k and β.

The posterior distribution on the parameters given the supervised outputs collected in y s is p(k, β|y s ) ∝ p(y s |k, β)p(k, β). Below we assume a uniform prior p(k, β) over the parameters. In this case MAP parameter estimation reduces to maximum marginal likelihood parameter estimation, i.e. find arg max k,β p(y s |k, β). Note that if the weights w ij are not fixed to 1/k but set to maximize the marginal likelihood, then a uniform prior on the parameters cannot be used. This is because the set of models with k + 1 neighbors then includes the set of models with k neighbors as a subset.

Since p is Gaussian, the marginal likelihood of the supervised points is obtained relatively easily. Using n s to denote the number of supervised points, the marginal log-likelihood of y s , for particular k and β, is given by:

log p(y s |β, k) = - 1 2 log |C ss | -n s log β + βy s C -1 ss y s , , (8)
where C ss depends on k. Through differentiation, the maximum of the marginal log-likelihood over β and the maximizer β * can be analytically obtained as:

β * = arg max β log p(y s |β, k) = n s /(y s C -1 ss y s ), (9)
which can be substituted in (8) to find

l * = log p(y s |β * , k) = - 1 2 log |C ss | + n s + n s log(y s C -1 ss y s /n s ) . (10)
Thus, to find the optimal number of neighbors k * we can evaluate [START_REF] Neal | Bayesian Learning for Neural Networks[END_REF] for k = {1, 2, . . . , k max } and pick the value for which [START_REF] Neal | Bayesian Learning for Neural Networks[END_REF] is maximum. To evaluate [START_REF] Neal | Bayesian Learning for Neural Networks[END_REF], we need to find C ss , which can be computed without explicitly computing all entries of C. Recall that C = M -1 , then using the Cholesky decomposition M = R R we can find the required entries of C ss by noting that

C ij = e i Ce j = e i (R R) -1 e j s i s j
, where e i is a vector with all zeros except for the i-th entry which equals 1, and s i = R -e i . Since R is triangular we can solve for s i in time linear in n.

Maximization of the marginal likelihood over the regularizer α cannot be performed in closed form, but it is possible to use gradient based techniques. The derivative of l * (the marginal loglikelihood maximized over β) with respect to the regularizer α equals:

∂l * ∂α = 1 2 Tr{C -1 ss [C 2 ] ss } - n s 2 y s C -1 ss [C 2 ] ss C -1 ss y s y s C -1 ss y s . ( 11
)
To find the required block of C 2 it is not necessary to explicitly compute this product. To see this, note that [C 2 ] ij = e i CCe j s i s j . Again using the Cholesky decomposition of M (ignoring β for clarity), we can write s i = Ce i = (R R) -1 e i . We can solve for s i by first solving R z = e i and then for Rs i = z. Thus, to find the elements of [C 2 ] ss we only need the Cholesky factor R and solve the triangular linear systems to find the required s i . However, after each gradient step the Cholesky factors will need to be recomputed. This makes optimization over the regularizer computationally demanding. In our experiments we therefore fixed the regularizer at α = 10 -11 .

Query selection

In a semi-supervised active learning setting a set of unlabelled points is given and the learning system asks the user to label one or more of these points to gain prediction accuracy on the remaining unlabelled points. The question is now: for a given labelled set of points, which data points should be additionally labelled to maximize prediction accuracy? To determine which query set of fixed size is the most informative, we may consider the entropy H(y u |y s ) of the conditional density of the unlabelled points given a labelled query set, and aim for the query set that leads to the lowest conditional entropy, see also [START_REF] Mackay | Bayesian Methods for Adaptive Models[END_REF][START_REF] Hwa | Sample selection for statistical grammar induction[END_REF]. Intuitively, we want the uncertainty of our predictions on the unlabelled points, given the labelled points, to be as small as possible. The rationale is that, if we assume that the mean of the field will approach the true value of the response variable, then we would like the distribution p(y u |y s ) to be concentrated as tight as possible around its mean. The chain-rule of entropies tells us that H(y) = H(y s ) + H(y u |y s ). Thus, since the entropy of the complete field H(y) is fixed, we can equivalently aim for the query set with the largest entropy. Note that that to compute the entropy H(y s ) = 1 2 log |C ss | -1 2 n s log β + const. we can use the Cholesky factors of M as above to find C ss .

To find the single variable with maximum variance we could compute all variances, the diagonal elements of C. However, already by selecting the maximum variance variable from a random subset of all variables, we have large probability of selecting one of the variables with largest variance. In fact, like in [START_REF] Smola | Sparse greedy matrix approximation for machine learning[END_REF], we can compute confidence bounds of the type: with probability at least 1-δ the variable with largest variance among m uniformly randomly selected variables is among the ε fraction of variables with largest variance, if m ≥ log δ/ log (1 -ε) . For example, if we use a random subset of 59 variables then with probability at least 95% the selected variable will be among the 5% of variables with the largest variances.

The number of possible query sets grows quickly with n s and heuristic search methods are inevitable already when n s is moderately large. A simple, and in practice very successful, scheme is to start with an initial random subset. Until some stopping criterion is met, randomly some variable y i not in the current subset is selected. An exchange is made if replacing one of the variables in the current query set with y i yields a higher joint entropy of the query set.

If we have some labelled points (the labels of which are collected in y t ) and want to find an additional set of queries indexed by s. Thus, the outputs are divided in two disjoint sets: the already labelled y t and the sofar unlabelled y u and the new queries are a subset of the sofar unlabelled : s ⊂ u. We can proceed as above, by considering the conditional field p(y u |y t ) and compute the required entropies from the Cholesky decomposition of M uu , rather than from the decomposition of M. (We can also use this to greedily, one-by-one, select an initial set of queries rather than randomly in the algorithm described above.) However, in an active learning system that computes a new query after the user has answered a previous query, the set of unlabelled points changes after each answered query. This would require to compute after each answered query the Cholesky factors of a different matrix M uu . 1 However, we can use the Cholesky factors of the whole matrix M to find a query set which maximally reduces the entropy in the distribution on the unlabelled points, if some points have already been labelled. To see this note that from the chain-rule of entropies we have that H(y u\s |y s∪t ) = H(y) -H(y s∪t ). Thus to find the next query set that maximally reduces the entropy in the remaining unlabelled points we find the query set s that, together with t, has the maximum entropy in the complete field. This can be done by evaluating the entropy H(y s∪t ) for different new query sets and picking the one that maximizes this entropy.

Experimental results

First we consider a face-pose estimation experiment to give a qualitative illustration of the techniques described above. The input data consists of a set of 2000 images of a face that looks in different directions, varying both horizontally and vertically (see the images depicted in Fig. 2). The goal is to predict the pose of the face.

A neighborhood graph with 20 neighbors was used that was computed directly on the basis of the squared distance between the 40 × 40 pixel images. The supervised data was obtained as follows: first the user has to place a random image in a two-dimensional plane. Then, using the minimum entropy principle a next query image is selected and the user places this image also in the plane. This process was repeated until ten images were placed in the plane. The two coordinates of the position in the plane of each image are considered as the supervised values of the two response variables which correspond to the horizontal and vertical viewing direction (pose) of the face.

In Fig. 2 the predicted pose parameters of all 2000 images are shown, the coordinates of the ten supervised images are indicated with the digits 1, . . . , 10. Observe that the query selection procedure mainly requested supervision for relatively extreme views. In the same figure we also plotted four lines, the images nearest to the circles on the lines are plotted next to the figure. These images show that using only 10 supervised examples a reasonable prediction on the pose depicted in the other images can be made.

We continue with quantitative experimental results obtained using a data set of 698 images (64×64 pixels) of a face rendered by computer graphics software. 2 The face is viewed and lit from different directions and for all images the ground truth viewing direction (longitude and latitude) and lighting direction (longitude) are available. The task is to predict the viewing direction and lighting direction of unsupervised images. The three response variables were normalized to the range [0, 3.5] yielding approximately unit variance. We fixed the regularizer α at 10 -11 .

In the experiments we used all 698 images. Some of the images were supervised and others not; the supervised images were selected either randomly or with by an active learning scheme. The squared errors between the predictions and the ground truth are averaged over the unsupervised points and over the different response variables. All reported results are averages over 20 experiments and we used t-tests to assess the p-value significance levels of the obtained results presented in Fig. 3 (note the log scale).

Query selection.

In Fig. 3 we show both for random query selection (dotted lines) and for active query selection (solid lines) the average squared errors over ten experiments against the number of neighbors k used in the model. The different lines (from top to bottom) show results for 10, 20 and 100 labelled points. We started the query selection procedure with the greedy query selection method, i.e. selecting one-by-one the queries that are expected to yield the most information given the other already selected query points. After the greedy initialization we improved the query set by an iterative procedure. We randomly selected one point not in the query set and computed whether the query set entropy increased if this point replaced any of the points in Errors on unsupervised points against number of neighbors, using 10, 20 and 100 randomly (dotted lines) and actively (solid lines) selected queries (top panel). Some example images (bottom right). Model selection results versus using optimal k value (table ).

the current query set. We terminated the iterations if no entropy increase was achieved after considering 20 unsupervised points in this manner.

We ran experiments for neighborhood graphs of different connectivity: k = 2, 3, . . . , 20. In all but 4 of the 57 combinations of k and a number of labelled points the active learning algorithm performs significantly (p < 0.001) better than using random queries. Also note that the standard deviation in the performance of the active learning approach is much smaller. We conclude that the active learning algorithm outperformed the usage of random queries.

Model selection.

Next, we consider the performance of the maximum likelihood model selection procedure described in Section 3.1. Note that query selection depends on the number of neighbors that are used, and that model selection depends on the supervised queries. To combine model selection with query selection we use a greedy procedure: first a random query is labelled, and then alternatingly k is selected using the labelled points (as described in Section 3.2), and a new query is selected depending on the current k and already labelled points (as described in Section 3.1). If instead a random query set is selected and labelled, we can do model selection directly using all labelled points.

The table in Fig. 3 shows the (average) results that were obtained with the optimal value of k in each experiment and with the value of k found by model selection. In each of the 20 experiments the optimal value of k was determined separately for using active and random queries: it is the number of neighbors for which the smallest prediction error was obtained (using active or random queries respectively).

For random queries using the optimal k does not give significantly better results than using model selection (p = 0.1). Thus model selection works well when using random queries. However, for actively selected queries and only 10 supervised points, the results for optimal k are significantly better (p < 0.02). The reason is that for few supervised points, model selection picks too small k which causes the query selection procedure to perform poorly. Comparing active and random queries, we found that for 10 queries there is no significant difference between active and random selection if model selection is used (p = 0.4). For more queries active selection is significantly better (p < 0.0001).

Correspondence learning

The second learning problem we consider is learning correspondences between two sets of data points. Each data set is assumed to be sampled from a different high-dimensional embedding of the same underlying low-dimensional manifold. The learning examples consist of (i) several 'supervised' pairs of points (one from each data set) that are in correspondence, i.e. have the same (unknown) coordinate on the underlying manifold and (ii) many 'unsupervised' points in each set for which the corresponding point in the other set is unknown. The goal is to predict other correspondences, i.e. for 'unsupervised' points we want to predict which point in the other set corresponds to it.

Correspondence learning with the LLE algorithm

The approach to correspondence learning taken in [START_REF] Ham | Learning high dimensional correspondences from low dimensional manifolds[END_REF] is to perform dimensionality reduction for both sets simultaneously with LLE, explicitly aiming at discovering the underlying lowdimensional coordinates. The energy function adds the separate LLE energy function (3) of each set. The given correspondences are accounted for by constraining the low-dimensional coordinates of corresponding points to coincide, as we explain next.

Without loss of generality we order the points in both sets in such a manner that the corresponding points have equal indices in each set, and all points with a correspondence have a smaller index than points without a correspondence. Let us use a and b to index the two data sets. Let L a and L b be the graph Laplacians, c.f. ( 1) and (3), of the separate sets as used by LLE. We partition L a as

L a = L a ss L a su L a us L a uu , (12)
and L b is partitioned accordingly. The vector y a is partitioned into a block for the points with correspondence y a s and a block for the other points y a u , and similarly for y b . Let

L =    L a ss + L b ss L a su L b su L a us L a uu 0 L b us 0 L b uu    . (13)
Taking into account the constraints y a s = y b s y s , and letting y = [y s y a u y b u ] , the energy function can be written as:

E(y) = y Ly. (14)
Analogous to the normal LLE algorithm, the eigenvectors of L with smallest eigenvalues give the minimum energy assignments of y a u , y b u , and y s . The low-dimensional coordinates of each point are in the corresponding row of the matrix of which the columns are the smallest eigenvectors of L. To predict a correspondence of an unsupervised point in set a, the nearest (using Euclidean distance in the low-dimensional representation) point from set b is found. Note that the dimensionality of the low-dimensional space, as well as relative scaling of the dimensions, influences the obtained results. In some cases an appropriate dimensionality for the embedding is known, in other cases however this dimensionality has to be estimated. Furthermore, in the above approach an appropriate number of nearest neighbors has to be determined for each set.

The semi-parametric correspondence learning setup in [START_REF] Verbeek | Non-linear CCA and PCA by Alignment of Local Models[END_REF] differs in that it delivers a mixture of factor analyzers from which the high-dimensional corresponding observations can be reconstructed without storing the training data. However, in the semi-parametric approach some trade-off has to be made between satisfying the correspondences (making sure that correspondences have similar low dimensional coordinates) and mutual alignment of the factor analyzers fitted to each set of observations (making sure that each factor analyzer assigns approximately the same low-dimensional coordinates to data points).

Correspondence learning with Gaussian fields

By casting the correspondence learning in the GF framework, we do not need an explicit representation in a low-dimensional space. Essentially, we compute quantities that are averaged over possible representations based on their likelihood under the GF distribution. As a first step we define a GF for each point set separately as in the previous sections, with inverse covariance matrices βM a and βM b . We may view this as a single GF with two independent subfields. To account for correspondences, we constrain variables that are linked by the correspondences to have identical (but unknown) values. The constraints induce correlations between other variables in the subfields. The inverse covariance matrix βM is obtained in the same way as L above, where the two variables of each correspondence pair have been replaced by a single variable. Again we use C to denote the inverse of M of the combined field. The expected squared difference (y i -y j ) 2 under the constrained GF distribution is:

E[(y i -y j ) 2 ] = E[y 2 i ] + E[y 2 j ] -2E[y i y j ] = β -1 (C ii + C jj -2C ij ). ( 15
)
Given a query point i from one set we can then find the point j in the other set with smallest expected squared difference (note that this point j is invariant for the value of β).

The LLE based method could be viewed as an approximation to our GF based method in the following sense. Let v t and λ t denote eigenvectors and the corresponding eigenvalues of C, then we can write

C ij = t v t (i)v t (j)λ t . (16)
Let the eigenvalues be ordered from large to small: λ t ≥ λ t+1 , the LLE based method [START_REF] Ham | Learning high dimensional correspondences from low dimensional manifolds[END_REF] is obtained (for embedding dimensionality d) as follows. We set λ t ← 1 for t ∈ {1, . . . , d + 1} and λ t ← 0 for t > d + 1. The squared distance in the eigenvector embedding used by the LLE based method equals C ii + C jj -2C ij , where the C ij are computed as in [START_REF] Verbeek | Non-linear CCA and PCA by Alignment of Local Models[END_REF] with the replaced λ t .

Model selection

In the correspondence problem we can use a marginal likelihood model selection procedure (selecting an appropriate number of neighbors k and variance scale parameter β) similar to the procedure used in the previous section. However, here we consider the high-dimensional input vectors as the variables that we want to predict. For the correspondences both high-dimensional input vectors belonging to sets a and b are known.

The model selection method of the previous section is based on maximizing the marginal log-likelihood of the supervised data. Suppose that the input vectors from set a and b have dimensionality respectively d a and d b . Let d = d a + d b , then we define the n s × d matrix Z as the matrix having in the i-th row the concatenation of the input vectors of the i-th correspondence pair. In the current setting, model selection will be based on maximizing the marginal loglikelihood of Z. Assuming all columns z j of Z are independently and identically distributed according to the GF distribution, this translates into maximizing the sum of the marginal loglikelihoods of the columns of Z. Thus model selection is based on finding k and β that maximize:

p(Z|k, β) = d j=1 log p(z j |k, β) (17)
= d j=1 - 1 2 log |C ss | -n s log β + βz j C -1 ss z j (18) = - 1 2 d log |C ss | -n s d log β + βTr{Z C -1 ss Z} . ( 19
)
The maximization with respect to β can again be done in closed form yielding an expressions similar to ( 9) and ( 10):

β * = arg max β log p(Z|β, k) = n s d/Tr{Z C -1 ss Z}, (20)
l * = log p(Z|β * , k) (21) = - 1 2 d log |C ss | + n s d + n s d log Tr{Z C -1 ss Z}/(n s d) . (22)
To find the value of k for which l * is maximal (and the corresponding β * ) we evaluate ( 22) for different values of k. To do this we need to find C ss , which can be computed with methods discussed in the previous section. Note that to compute the trace in (22) we can use Tr{Z C -1 ss Z} = d j=1 z j C -1 ss z j . Thus the required amount of computation grows only linearly in the number of input dimensions d.

Experimental results

We compare the performance of our GF approach with the LLE approach proposed in [START_REF] Ham | Learning high dimensional correspondences from low dimensional manifolds[END_REF] using representations of different dimensionality. The data sets we used each contain 2500 views of a toy puppet recorded by positioning a camera on different positions on the hemisphere centered at the object. 3 For each view of one object there is a corresponding view of the other object recorded from the same camera position, see the images in Fig. 4. Given some corresponding views, the task is to predict for each remaining image in set a which is the corresponding image in set b. We compare the position of the camera of the true correspondence and the predicted correspondence. The error we measure is the summed squared error in the longitude of camera (0 • -360 • ) and latitude (0 • -90 • ). The neighborhood graph (using k = 20 neighbors) was based on Euclidean distances when between the 64 × 64 pixel images (considering each pixel as a coordinate in the image space).

In Fig. 4 the mean and standard deviation of the errors are shown of ten experiments using different random sets of 20, 50, 100 and 200 correspondences. First, we can see that for dimensionality smaller than three the LLE based method performs poorly; the differences with GF are significant with p < 0.01. The reason is that of the two degrees of freedom in the data, one (the longitude of the camera) is periodic. Therefore an additional dimension is required for a suitable representation. Second, the GF approach leads to equal or smaller errors and has less variance in the errors than the LLE based method using three or more dimensions and less than 100 correspondences (7 of the 9 differences are significant with p < 0.05). We conclude that the GF method (i) removes the need to find the optimal representation dimensionality and (ii) performs as good as or better than the LLE based method. In Fig. 5 we show results for model selection, averaged over ten experiments, using different randomly selected sets of 50 correspondences. In each experiment we computed for k = 2, 4, 6, . . . , 20 neighbors the marginal log-likelihood l * of the 2 × 64 2 observed pixel values of the correspondences. In each experiment, we also predicted the correspondences with the LLE based method and the GF method. The average prediction error and marginal loglikelihood are plotted against the number of neighbors in Fig. 5. Note that marginal likelihood gives a good indication of the prediction error and that -on average -for k = 8 the highest marginal log-likelihood is obtained and both methods give the smallest prediction errors. Another interesting quantity is the average prediction error when using in each experiment the value of k for which the maximum marginal log-likelihood was achieved in that experiment. This is the error that would be obtained if we used the maximum likelihood criterion for model selection. The average of this error is given by the horizontal lines in the left panel of Fig. 5. In five experiments k = 8 was selected, three times k = 10 was selected and in two cases k = 12 was selected.

Conclusion and discussion

In this paper we have shown how sparse GFs based on a neighborhood graph can be used for semi-supervised regression and correspondence learning. For semi-supervised regression we have shown that (i) a number of neighbors that yields good generalization can effectively be estimated using a maximum-likelihood criterion and (ii) smaller errors can be obtained with an equal number of labeled examples if an entropy minimization principle is used to select which data points to label. For correspondence learning we have shown that our GF approach, for which we do not have to fix a representation dimensionality, performs comparable or better than the LLE based method. Furthermore, also for correspondence learning a number of neighbors that gives accurate predictions can be determined through a maximum likelihood criterion.

An interesting alternative to using Cholesky decompositions, is to use iterative algorithms to solve linear systems, e.g. methods based on Lanczos iterations [START_REF] Golub | Matrix computations[END_REF], to find the blocks of C and C -1 required for model selection and query selection. Such methods are attractive in active learning settings since the solution after labelling a new point will be similar to the solution before labelling the point. Therefore, accurate solutions can be found in relatively few iterations if the iterations are started with the (approximate) solution before the last labelling [START_REF] Mahdaviani | Fast computational methods for visually guided robots[END_REF].

The Gaussian field (GF) models considered here resemble Gaussian Processes (GP) [START_REF] Neal | Bayesian Learning for Neural Networks[END_REF]. A GP is given by a mean function m(x) and a covariance function C(x, x ) which together define a Gaussian density over the outputs for all points in some input domain. The marginal Gaussian on the outputs of n elements x 1 , x 2 , . . . x n of the input domain can be obtained as follows: the mean vector is given by [m(x 1 ) . . . m(x n )] and the (i, j) th entry of the covariance matrix is given by C(x i , x j ). Thus, the data are used to specify the covariance matrix rather than the inverse covariance function as in the GF considered in this paper. Therefore, in a GP the marginal on the outputs of several data points does not depend on other unsupervised data points. The marginal is simply obtained by removing the entries in the mean and covariance matrix that correspond to the variables over which we we want to marginalize. The important difference between GP and the GF considered here is that for the GF the marginal distribution on the outputs of several data points does depend on other unsupervised points (for a fixed matrix M). The covariance matrix of the marginal is given by the corresponding block of the inverse of M and thus depends also on the parts of M corresponding to unsupervised points. In this manner unsupervised data helps to identify the manifold structure of the data. The manifold structure is reflected in the covariance matrix, the entries of which are not simply given by (a transformation of) the Euclidean distance between data points, but also depend on other data points that indicate how distant the points are on the manifold.

Semi-supervised learning is a machine learning approach that is very relevant when working with high-dimensional data sets (e.g. images or text); in particular for nearest neighbor based methods, due to their flexible non-parametric nature. Unfortunately nearest neighbor methods need to access the training data to make predictions for future data. In future research we want to consider combinations of nearest-neighbor models and parametric models. In particular we want to use semi-supervised GF techniques (like those described here and in [START_REF] Zhu | Semi-supervised learning: From gaussian fields to gaussian processes[END_REF]) to define a prior on the partially observed variable of interest y (e.g. a class label or response variable). This prior can then be used to bias parametric model learning algorithms, like the latent variable model proposed in [START_REF] Roweis | Global coordination of local linear models[END_REF], towards models that comply with the GF distribution on y.

Figure 1 :

 1 Figure 1: Input data distributed along spiral (left panel). True and predicted output for the data (right panel). The supervised points are encircled in both panels. Predictions are made using the direct and LLE based method to specify matrix A.

Figure 2 :

 2 Figure 2: Predicted pose parameters of 2000 images, together with the pose parameters of the 10 supervised images. For 40 other points in the pose space we plotted next to the figure the image with the most similar predicted pose-parameters.

  : random / active queries O/S: using optimal/selected k

Figure 3 :

 3 Figure3: Errors on unsupervised points against number of neighbors, using 10, 20 and 100 randomly (dotted lines) and actively (solid lines) selected queries (top panel). Some example images (bottom right). Model selection results versus using optimal k value (table).

Figure 4 :

 4 Figure 4: Mean (top left) and standard deviations (top right) of error against number of correspondences. Results are shown for the LLE based method using 1 up to 5 dimensions and the GF method. Three examples of corresponding images connected by bars (bottom).

Figure 5 :

 5 Figure 5: Mean error against the number of neighbors, for LLE based method (dotted) and GF (solid). Horizontal lines give the average obtained result if model selection is used (left panel). Mean and standard deviations of marginal log-likelihood against the number of neighbors (right panel).

Note that also to predict y u the Cholesky factor of Muu has to be computed.

The data, in[START_REF] Tenenbaum | A global geometric framework for nonlinear dimensionality reduction[END_REF] used for unsupervised learning, is available from http://isomap.stanford.edu.

The data is available through G. Peters (Dept. of Computer Graphics, University Dortmund, Germany), see[START_REF] Peters | How to measure the pose robustness of object views[END_REF] and http://ls7-www.cs.uni-dortmund.de/~peters.

Acknowledgements

JJV is supported by the Technology Foundation STW (AIF4997) applied science division of NWO and the technology program of the Dutch Ministry of Economic Affairs.