Convexity, Classification, and Risk Bounds

Many of the classification algorithms developed in the machine learning literature, including the support vector machine and boosting, can be viewed as minimum contrast methods that minimize a convex surrogate of the 0–1 loss function. The convexity makes these algorithms computationally efficient. The use of a surrogate, however, has statistical consequences that must be balanced against the computational virtues of convexity. To study these issues, we provide a general quantitative relationship between the risk as assessed using the 0–1 loss and the risk as assessed using any nonnegative surrogate loss function. We show that this relationship gives nontrivial upper bounds on excess risk under the weakest possible condition on the loss function—that it satisfies a pointwise form of Fisher consistency for classification. The relationship is based on a simple variational transformation of the loss function that is easy to compute in many applications. We also present a refined version of this result in the case of low noise, and show that in this case, strictly convex loss functions lead to faster rates of convergence of the risk than would be implied by standard uniform convergence arguments. Finally, we present applications of our results to the estimation of convergence rates in function classes that are scaled convex hulls of a finite-dimensional base class, with a variety of commonly used loss functions.

Correction to “The Importance of Convexity in Learning With Squared Loss” [Sep 98 1974-1980]

The paper "the importance of convexity in learning with squared loss" gave a lower bound on the sample complexity of learning with quadratic loss using a nonconvex function class. The proof contains an error. We show that the lower bound is true under a stronger condition that holds for many cases of interest.

Total factor productivity growth and convergence in the petroleum industry : empirical analysis testing for convexity

While economic theory acknowledges that some features of technology (e.g., indivisibilities, economies of scale and specialization) can fundamentally violate the traditional convexity assumption, almost all empirical studies accept the convexity property on faith. In this contribution, we apply two alternative flexible production technologies to measure total factor productivity growth and test the significance of the convexity axiom using a nonparametric test of closeness between unknown distributions. Based on unique field level data on the petroleum industry, the empirical results reveal significant differences, indicating that this production technology is most likely non-convex. Furthermore, we also show the impact of convexity on answers to traditional convergence questions in the productivity growth literature.

Classification with a reject option using a hinge loss

We consider the problem of binary classification where the classifier can, for a particular cost, choose not to classify an observation. Just as in the conventional classification problem, minimization of the sample average of the cost is a difficult optimization problem. As an alternative, we propose the optimization of a certain convex loss function φ, analogous to the hinge loss used in support vector machines (SVMs). Its convexity ensures that the sample average of this surrogate loss can be efficiently minimized. We study its statistical properties. We show that minimizing the expected surrogate loss—the φ-risk—also minimizes the risk. We also study the rate at which the φ-risk approaches its minimum value. We show that fast rates are possible when the conditional probability P(Y=1|X) is unlikely to be close to certain critical values.

Segmental torso masses and coronal plane joint torques in the adolescent scoliotic spine

Introduction. Calculating segmental (vertebral level-by-level) torso masses in Adolescent Idiopathic Scoliosis (AIS) patients allows the gravitational loading on the scoliotic spine during relaxed standing to be determined. This study used CT scans of AIS patients to measure segmental torso masses and explores how joint moments in the coronal plane are affected by changes in the position of the intervertebral joint’s axis of rotation; particularly at the apex of a scoliotic major curve. Methods. Existing low dose CT data from the Paediatric Spine Research Group was used to calculate vertebral level-by-level torso masses and joint torques occurring in the spine for a group of 20 female AIS patients (mean age 15.0 ± 2.7 years, mean Cobb angle 53 ± 7.1°). Image processing software, ImageJ (v1.45 NIH USA) was used to threshold the T1 to L5 CT images and calculate the segmental torso volume and mass corresponding to each vertebral level. Body segment masses for the head, neck and arms were taken from published anthropometric data. Intervertebral (IV) joint torques at each vertebral level were found using principles of static equilibrium together with the segmental body mass data. Summing the torque contributions for each level above the required joint, allowed the cumulative joint torque at a particular level to be found. Since there is some uncertainty in the position of the coronal plane Instantaneous Axis of Rotation (IAR) for scoliosis patients, it was assumed the IAR was located in the centre of the IV disc. A sensitivity analysis was performed to see what effect the IAR had on the joint torques by moving it laterally 10mm in both directions. Results. The magnitude of the torso masses from T1-L5 increased inferiorly, with a 150% increase in mean segmental torso mass from 0.6kg at T1 to 1.5kg at L5. The magnitudes of the calculated coronal plane joint torques during relaxed standing were typically 5-7 Nm at the apex of the curve, with the highest apex joint torque of 7Nm being found in patient 13. Shifting the assumed IAR by 10mm towards the convexity of the spine, increased the joint torque at that level by a mean 9.0%, showing that calculated joint torques were moderately sensitive to the assumed IAR location. When the IAR midline position was moved 10mm away from the convexity of the spine, the joint torque reduced by a mean 8.9%. Conclusion. Coronal plane joint torques as high as 7Nm can occur during relaxed standing in scoliosis patients, which may help to explain the mechanics of AIS progression. This study provides new anthropometric reference data on vertebral level-by-level torso mass in AIS patients which will be useful for biomechanical models of scoliosis progression and treatment. However, the CT scans were performed in supine (no gravitational load on spine) and curve magnitudes are known to be smaller than those measured in standing.

Traditional and geometric morphometrics detect morphological variation of lower pharyngeal jaw inCoris julis(Teleostei, Labridae)

In the present study, variation in the morphology of the lower pharyngeal element between two Sicilian populations of the rainbow wrasse Coris julis has been explored by the means of traditional morphometrics for size and geometric morphometrics for shape. Despite close geographical distance and probable high genetic flow between the populations, statistically significant differences have been found both for size and shape. In fact, one population shows a larger lower pharyngeal element that has a larger central tooth. Compared to the other population, this population also has medially enlarged lower pharyngeal jaws with a more pronounced convexity of the medial-posterior margin. The results are discussed in the light of a possible more pronounced durophagy of this population.

Morphometric analysis of the thoracic intervertebral foramen osseous anatomy in adolescent idiopathic scoliosis using low dose computerized tomography

Introduction. The dimensions of the thoracic intervertebral foramen in adolescent idiopathic scoliosis (AIS) have not previously been quantified. During posterior approach scoliosis correction surgery pedicle screws may occasionally breach into the foramen. Better understanding of the dimensions of the foramen may be useful in surgical planning. This study describes a reproducible method for measurement of the thoracic foramen in AIS using computerized tomography (CT). Methods. In 23 pre-operative female patients with Lenke 1 type AIS with right side convexity major curves confined to the thoracic spine the foraminal height (FH), foraminal width (FW), pedicle to superior articular process distance (P-SAP) and cross sectional foraminal area (FA) were measured using multiplanar reconstructed CT. Measurements were made at entrance, midpoint and exit of the thoracic foramina from T1/T2 to T11/T12. Results were correlated with potential dependent variables of major curve Cobb Angle measured on X-ray and CT, Age, Weight, Lenke classification subtype, Risser Grade and number of spinal levels in the major curve. Results. The FH, FW, P-SAP and FA dimensions and ratios are all significantly larger on the convexity of the major curve and maximal at or close to the apex. Mean thoracic foraminal dimensions change in a predictable manner relative to position on the major thoracic curve. There was no significant correlation with the measured foraminal dimensions or ratios and the potential dependent variables. The average ratio of convexity to concavity dimensions at the apex foramina for entrance, midpoint and exit respectively are FH (1.50, 1.38, 1.25), FW (1.28, 1.30, 0.98), FA (2.06, 1.84, 1.32), P-SAP (1.61, 1.47, 1.30). Conclusion. Foraminal dimensions of the thoracic spine are significantly affected by AIS. Foraminal dimensions have a predictable convexity to concavity ratio relative to the proximity to the major curve apex. Surgeons should be aware of these anatomical differences during scoliosis correction surgery.

Using low dose computerized tomography to morphometrically analyse the thoracic intervertebral foramen osseous anatomy in adolescent idiopathic scoliosis

INTRODUCTION The dimensions of the thoracic intervertebral foramen in adolescent idiopathic scoliosis (AIS) have not previously been quantified. During posterior approach scoliosis correction surgery pedicle screws may occasionally breach into the foramen. Better understanding of the dimensions of the foramen may be useful in surgical planning. This study describes a reproducible method for measurement of the thoracic foramen in AIS using computerized tomography (CT). METHODS In 23 pre-operative female patients with Lenke 1 type AIS with right side convexity major curves confined to the thoracic spine the foraminal height (FH), foraminal width (FW), pedicle to superior articular process distance (P-SAP) and cross sectional foraminal area (FA) were measured using multiplanar reconstructed CT. Measurements were made at entrance, midpoint and exit of the thoracic foramina from T1/T2 to T11/T12. Results were correlated with potential dependent variables of major curve Cobb Angle measured on X-ray and CT, Age, Weight, Lenke classification subtype, Risser Grade and number of spinal levels in the major curve. RESULTS The FH, FW, P-SAP and FA dimensions and ratios are all significantly larger on the convexity of the major curve and maximal at or close to the apex. Mean thoracic foraminal dimensions change in a predictable manner relative to position on the major thoracic curve. There was no significant correlation with the measured foraminal dimensions or ratios and the potential dependent variables. The average ratio of convexity to concavity dimensions at the apex foramina for entrance, midpoint and exit respectively are FH (1.50, 1.38, 1.25), FW (1.28, 1.30, 0.98), FA (2.06, 1.84, 1.32), P-SAP (1.61, 1.47, 1.30). CONCLUSION Foraminal dimensions of the thoracic spine are significantly affected by AIS. Foraminal dimensions have a predictable convexity to concavity ratio relative to the proximity to the major curve apex. Surgeons should be aware of these anatomical differences during scoliosis correction surgery.