On the Number of Linear Regions of Convolutional Neural Networks With Piecewise Linear Activations.

One fundamental problem in deep learning is understanding the excellent performance of deep Neural Networks (NNs) in practice. An explanation for the superiority of NNs is that they can realize a large family of complicated functions, i.e., they have powerful expressivity. The expressivity of a Neural Network with Piecewise Linear activations (PLNN) can be quantified by the maximal number of linear regions it can separate its input space into. In this paper, we provide several mathematical results needed for studying the linear regions of Convolutional Neural Networks with Piecewise Linear activations (PLCNNs), and use them to derive the maximal and average numbers of linear regions for one-layer PLCNNs. Furthermore, we obtain upper and lower bounds for the number of linear regions of multi-layer PLCNNs. Our results suggest that deeper PLCNNs have more powerful expressivity than shallow PLCNNs, while PLCNNs have more expressivity than fully-connected PLNNs per parameter, in terms of the number of linear regions.

Gender differences and other factors associated with HIV testing in a national sample of active drug injectors.

Gender, health, HIV risk, and demographic factors were examined with chi-square and logistic regression analyses to assess which of these factors are most predictive of active injection drug users' (IDUs') getting tested for HIV. Analyses were based on 36,898 IDUs recruited to participate in a nationwide multisite HIV prevention project. Women IDUs were recently tested (prior 6 months) more than men IDUs. Health factors, particularly for women IDUs, predicted who got tested for HIV more than risk or demographic factors. HIV testing usually occurred when disease symptomatology developed, after a long period of being capable of transmitting HIV to others, and when the benefits of AZT and other interventions may be reduced. Preventive interventions with IDUs are needed that emphasize the possible benefits of HIV testing and that encourage testing before symptoms develop.

HIV/AIDS-related behavior change among injecting drug users in different national settings.

OBJECTIVES: To identify factors associated with effective AIDS behavior change among injecting drug users (IDU) in different national settings. DESIGN: Cross-sectional surveys of IDU, with determination of HIV status. Trends in city HIV seroprevalence among IDU also used to validate effectiveness of behavior change. SETTING AND PARTICIPANTS: Subjects recruited from drug-use treatment programs and outreach efforts in Bangkok, Thailand (n = 601), Glasgow, Scotland (n = 919), New York City, USA (n = 2539), and Rio de Janeiro, Brazil (n = 466). RESULTS: Evidence for the effectiveness of self-reported risk reduction was available for all cities. Univariate followed by multiple logistic regression analyses were used to identify factors associated with self-reported AIDS behavior change. Separate analyses were conducted for each city. Talking about AIDS with drug-using friends was significantly associated with behavior change in all four cities. Talking with sex partners about AIDS, educational level, knowing that someone can be HIV-infected and still look healthy, and having been tested previously for HIV were each significantly associated with behavior change in three of the four cities. CONCLUSIONS: Despite the substantial differences in these national settings, there were common factors associated with effective risk reduction. In particular, risk reduction appears to occur through social processes rather than through individual attitude change. HIV prevention programs need to explicitly incorporate social processes into their work.

Continuity and change within an HIV epidemic. Injecting drug users in New York City, 1984 through 1992.

OBJECTIVES: To examine trends in acquired immunodeficiency syndrome (AIDS) risk behavior and human immunodeficiency virus (HIV) seroprevalence among injecting drug users (IDUs) in New York City from 1984 through 1992. DESIGN AND SETTING: Comparisons were made between two surveys of IDUs at the same hospital-based New York City drug abuse detoxification program: 141 IDUs in 1984 and 974 IDUs in 1990 through 1992. National Death Registry, New York City Health Department, and drug treatment program records were also used. PARTICIPANTS: Persons attending detoxification program randomly selected for participation. Eligibility was based on injection within previous 2 months; 99% acceptance rates were obtained. Participants in the 1984 and 1990 through 1992 surveys were 66% and 79% men, 21% and 19% white, 33% and 34% African American, and 45% and 46% Latin American, respectively. INTERVENTIONS: Community-based AIDS prevention programs, including underground syringe exchanges. MAIN OUTCOME MEASURES: Acquired immunodeficiency syndrome risk behaviors; HIV serostatus; CD4+ cell counts; death rates among 1984 subjects; and injection and intranasal routes of drug administration. RESULTS: The HIV seroprevalence remained stable at slightly more than 50%. Mean CD4+ cell counts declined from 0.716 x 10(9)/L (716/microL) to 0.575 x 10(9)/L (P < .009). Annual death rate among 1984 subjects was 3%, with a significantly higher rate among HIV-seropositive subjects (relative risk, 2.57; 95% exact binomial confidence interval, 1.12 to 6.61). Large-scale declines were observed in AIDS risk behaviors, eg, use of potentially contaminated syringes declined from 51% to 7% of injections (P < .001). Recent additional risk reduction was associated with use of the underground syringe exchanges. Intranasal heroin use was the primary route of drug administration for 46% of heroin admissions to New York City drug treatment programs. CONCLUSIONS: The HIV seroprevalence has remained stable among this population of New York City IDUs for almost a decade. Continuation of current trends should lead to further reduction in HIV transmission, although reversal of the trend to intranasal use could lead to substantially increased transmission.

CD4 lymphocytopenia among injecting drug users in New York City.

Recent cases of "AIDS-like" CD4 lymphocytopenia in the absence of HIV infection have generated considerable scientific and public interest. We studied CD4 cell counts and percentages from 1984 to 1992 among 1,246 HIV-seronegative injecting drug users in New York City, a population at very high risk for exposure to bloodborne pathogens. Severe CD4 lymphocytopenia was rare, and there was no evidence of an increase over time. Of 229 subjects with longitudinal data, only four met the surveillance definition for "idiopathic CD4 lymphocytopenia" (ICL). CD4 cell counts of < 500 cells/microliters were, however, associated with subsequent HIV seroconversion (12.7/100 person-years at risk, relative risk (RR) = 4.53, 95% exact binomial confidence interval (CI) 1.7-10.7, p = 0.002).

Condom use with primary partners among injecting drug users in Bangkok, Thailand and New York City, United States.

OBJECTIVE: To determine factors associated with likelihood or failure to use condoms with primary sexual partners among injecting drug users (IDU) in two cities. DESIGN AND METHODS: Interviews were conducted with 601 IDU in Bangkok in 1989 and with 957 IDU in New York City in 1990-1991. Subjects were recruited from drug-use treatment programs and a research storefront. Informed consent was obtained and a World Health Organization standardized questionnaire about AIDS risk behaviors administered by a trained interviewer. RESULTS: A substantial minority (37%) of IDU in Bangkok and a majority (55%) of IDU in New York City reported penetrative intercourse (vaginal, anal or oral) with a primary partner in the 6 months before the interview. Of those reporting penetrative intercourse with a primary partner, only 12% in Bangkok and 20% in New York reported that they always used condoms. Parallel bivariate and multiple logistic regression analyses were conducted to distinguish between subjects who reported always using condoms and subjects who reported unsafe sexual activity with primary partners. The same two factor--knowing that one is HIV-seropositive and talking about AIDS with sexual partners--were most strongly associated with always using condoms with primary partners in both cities. CONCLUSIONS: Programs to prevent sexual transmission of HIV among IDU should provide voluntary and confidential/anonymous HIV counseling and testing, and should facilitate discussions of AIDS and sexual transmission of HIV between IDU and their sexual partners. That the same two factors were associated with always using condoms with primary partners among IDU in these two cities suggests that these factors may also be important in other groups at high risk for HIV.

PIP: Trained interviewers spoke to 957 drug users attending a detoxification program, methadone maintenance program, or a research storefront in New York City in 1990-91 and to 601 drug users attending 17 drug use treatment clinics in Bangkok, Thailand, in the autumn of 1989 as part of a study to identify factors linked to the probability or failure of condom use with primary sexual partners among IV drug users. The participants also received HIV counseling and testing. IV drug users in New York City were more likely to be older (36.2 years vs. 30.1 years; p .001), female (25% vs. 5%; p .001), more ethnically diverse (p .001), and inject cocaine more often (33 injections/month vs. 0.5 injections/month) than those in Bangkok. 44% of drug users in New York City and 33% of those in Bangkok engaged in some unprotected penetrative intercourse with a primary heterosexual partner in the previous 6 months. Of drug users having penetrative sexual intercourse with a primary partner in the previous 6 months, 20% in New York City and 12% in Bangkok always used condoms (p .02). The strongest predictors of condom use among IV drug users from both countries were a previous positive HIV test and talking about AIDS with sexual partners (p = .001 for US; p = .0008 for Bangkok and p = .004 for US; p = .0596 for Bangkok, respectively). These findings indicated that unsafe sexual behavior with primary sexual partners among drug users is still a major source of HIV transmission in these 2 cities. Nevertheless, knowledge of HIV positive status and partner communication concerning AIDS are predictors of condom use shared by both groups. Thus, HIV/AIDS prevention programs should provide confidential HIV testing and counseling for drug users and should encourage frank discussions of AIDS between drug users and primary sexual partners. Peer support for risk reduction among drug users has the potential to facilitate such discussions.

Implications of the revised surveillance definition: AIDS among New York City drug users.

The Centers for Disease Control (CDC) has proposed revising the AIDS surveillance definition to include any HIV-seropositive person with a CD4 cell count of less than 200 cells per microliter. Based on a study of persons receiving treatment for HIV infection, this new definition would lead to an estimated 50% increase in the number of persons recognized as living with AIDS. Among 440 HIV-seropositive research subjects recruited from drug treatment programs and through street outreach in New York City, 59 met this definition, yet only 25% of those had been reported to the New York City AIDS registry. The new definition, if combined with HIV and T-cell testing at drug treatment and street outreach programs, could thus yield very large increases in the number of injecting drug users meeting the new surveillance definition of AIDS.

Crack cocaine use in a cohort of methadone maintenance patients.

We examined crack use in a cohort of methadone patients originally enrolled in 1984-86. Crack use questions were added to the study in 1987. Of the 494 methadone patients originally enrolled, 228 subjects remained in methadone and were re-interviewed in 1987-88, and 234 remained in methadone and were re-interviewed in 1988-89. Approximately one-quarter of the subjects were using crack at each of the 1987-88 and 1988-89 data collection points, and only 3% of the subjects were using crack at daily or greater frequencies at each of the 1987-88 and 1988-89 interviews. Concurrent crack use was associated with (a) the number of noninjected drugs being used; (b) the number of IV drug-using sexual partners; (c) drug injection; and (d) the use of nonheroin opiates. Persistent crack use, defined as use in both 1987-88 and 1988-89, was associated with previous noninjected drug use and previous suicide attempts. While the potential problem of crack use among methadone patients should not be minimized, it appears that, compared to illicit drug injectors not in treatment, being in methadone maintenance may offer a protective effect against crack use.

Controlled study of the impact of educational home visits by pharmacists to high-risk older patients.

Lack of information about medications coupled with high rates of utilization complicates compliance with medication regimens and increases the risk of adverse effects among older adults. We undertook a study of the efficacy of community-based interventions by pharmacists in a randomly-allocated one-half of a sample of 284 older adults considered to be at high risk for medication-related problems. Information and attitudes towards prescription and over-the-counter medications did not differ significantly between the intervention and comparison groups, either before or after the pharmacist interventions. However, visits to physicians were significantly less in the intervention group, suggesting an important if unexpected impact on health-related behavior.

Sleep problems in the community elderly as predictors of death and nursing home placement.

In 1984-85, 1855 elderly residents of an urban community responded to a comprehensive baseline interview that included questions regarding an extensive set of sleep characteristics and problems. During the subsequent 3 1/2 years of follow-up, 16.7% of the respondents died and 3.5% were placed in nursing homes. The predictive significance of each sleep characteristic for mortality and for nursing home placement was determined separately for males and females, using Cox proportional hazards models. Selected demographic and psychosocial variables were also entered into the models. Age, problems with activities of daily living (ADL), self-assessed health, income, cognitive impairment, depression and whether respondents were living alone were controlled for statistically. Of the many variables analyzed, in males insomnia was the strongest predictor of both mortality and nursing home placement. For mortality, the relative hazard associated with insomnia exceeded the hazards associated with age, ADL problems, fair-poor health and low income. For nursing home placement, the hazard associated with insomnia exceeded that associated with cognitive impairment. The relationships of insomnia to mortality and nursing home placement were U-shaped, with a worse outcome if insomnia complaints over the preceding 2 weeks were either prominent (numerous or frequent) or absent. For females, insomnia was a borderline predictor of mortality and did not predict nursing home placement at all. Symptoms of the restless legs syndrome predicted mortality for females in some Cox regression models. Reported sleep duration, symptoms of sleep apnea and frequent use of hypnotic drugs did not predict mortality or nursing home placement in either sex.