Best rank-1 approximations without orthogonal invariance for the 1-norm

Data measured in the real-world is often composed of both a true signal, such as an image or experimental response, and a perturbation, such as noise or weak secondary effects. Low-rank matrix approximation is one commonly used technique to extract the true signal from the data. Given a matrix representation of the data, this method seeks the nearest low-rank matrix where the distance is measured using a matrix norm. The classic Eckart-Young-Mirsky theorem tells us how to use the Singular Value Decomposition (SVD) to compute a best low-rank approximation of a matrix for any orthogonally invariant norm. This leaves as an open question how to compute a best low-rank approximation for norms that are not orthogonally invariant, like the 1-norm. In this thesis, we present how to calculate the best rank-1 approximations for 2-by-n and n-by-2 matrices in the 1-norm. We consider both the operator induced 1-norm (maximum column 1-norm) and the Frobenius 1-norm (sum of absolute values over the matrix). We present some thoughts on how to extend the arguments to larger matrices

A molecularly integrated grade for meningioma.

BackgroundMeningiomas are the most common primary intracranial tumor in adults. Clinical care is currently guided by the World Health Organization (WHO) grade assigned to meningiomas, a 3-tiered grading system based on histopathology features, as well as extent of surgical resection. Clinical behavior, however, often fails to conform to the WHO grade. Additional prognostic information is needed to optimize patient management.MethodsWe evaluated whether chromosomal copy-number data improved prediction of time-to-recurrence for patients with meningioma who were treated with surgery, relative to the WHO schema. The models were developed using Cox proportional hazards, random survival forest, and gradient boosting in a discovery cohort of 527 meningioma patients and validated in 2 independent cohorts of 172 meningioma patients characterized by orthogonal genomic platforms.ResultsWe developed a 3-tiered grading scheme (Integrated Grades 1-3), which incorporated mitotic count and loss of chromosome 1p, 3p, 4, 6, 10, 14q, 18, 19, or CDKN2A. 32% of meningiomas reclassified to either a lower-risk or higher-risk Integrated Grade compared to their assigned WHO grade. The Integrated Grade more accurately identified meningioma patients at risk for recurrence, relative to the WHO grade, as determined by time-dependent area under the curve, average precision, and the Brier score.ConclusionWe propose a molecularly integrated grading scheme for meningiomas that significantly improves upon the current WHO grading system in prediction of progression-free survival. This framework can be broadly adopted by clinicians with relative ease using widely available genomic technologies and presents an advance in the care of meningioma patients

Targeted gene expression profiling predicts meningioma outcomes and radiotherapy responses

Surgery is the mainstay of treatment for meningioma, the most common primary intracranial tumor, but improvements in meningioma risk stratification are needed and indications for postoperative radiotherapy are controversial. Here we develop a targeted gene expression biomarker that predicts meningioma outcomes and radiotherapy responses. Using a discovery cohort of 173 meningiomas, we developed a 34-gene expression risk score and performed clinical and analytical validation of this biomarker on independent meningiomas from 12 institutions across 3 continents (N = 1,856), including 103 meningiomas from a prospective clinical trial. The gene expression biomarker improved discrimination of outcomes compared with all other systems tested (N = 9) in the clinical validation cohort for local recurrence (5-year area under the curve (AUC) 0.81) and overall survival (5-year AUC 0.80). The increase in AUC compared with the standard of care, World Health Organization 2021 grade, was 0.11 for local recurrence (95% confidence interval 0.07 to 0.17, P &lt; 0.001). The gene expression biomarker identified meningiomas benefiting from postoperative radiotherapy (hazard ratio 0.54, 95% confidence interval 0.37 to 0.78, P = 0.0001) and suggested postoperative management could be refined for 29.8% of patients. In sum, our results identify a targeted gene expression biomarker that improves discrimination of meningioma outcomes, including prediction of postoperative radiotherapy responses.</p

Targeted gene expression profiling predicts meningioma outcomes and radiotherapy responses

Surgery is the mainstay of treatment for meningioma, the most common primary intracranial tumor, but improvements in meningioma risk stratification are needed and indications for postoperative radiotherapy are controversial. Here we develop a targeted gene expression biomarker that predicts meningioma outcomes and radiotherapy responses. Using a discovery cohort of 173 meningiomas, we developed a 34-gene expression risk score and performed clinical and analytical validation of this biomarker on independent meningiomas from 12 institutions across 3 continents (N = 1,856), including 103 meningiomas from a prospective clinical trial. The gene expression biomarker improved discrimination of outcomes compared with all other systems tested (N = 9) in the clinical validation cohort for local recurrence (5-year area under the curve (AUC) 0.81) and overall survival (5-year AUC 0.80). The increase in AUC compared with the standard of care, World Health Organization 2021 grade, was 0.11 for local recurrence (95% confidence interval 0.07 to 0.17, P &lt; 0.001). The gene expression biomarker identified meningiomas benefiting from postoperative radiotherapy (hazard ratio 0.54, 95% confidence interval 0.37 to 0.78, P = 0.0001) and suggested postoperative management could be refined for 29.8% of patients. In sum, our results identify a targeted gene expression biomarker that improves discrimination of meningioma outcomes, including prediction of postoperative radiotherapy responses.</p

Targeted gene expression profiling predicts meningioma outcomes and radiotherapy responses

Surgery is the mainstay of treatment for meningioma, the most common primary intracranial tumor, but improvements in meningioma risk stratification are needed and indications for postoperative radiotherapy are controversial. Here we develop a targeted gene expression biomarker that predicts meningioma outcomes and radiotherapy responses. Using a discovery cohort of 173 meningiomas, we developed a 34-gene expression risk score and performed clinical and analytical validation of this biomarker on independent meningiomas from 12 institutions across 3 continents (N = 1,856), including 103 meningiomas from a prospective clinical trial. The gene expression biomarker improved discrimination of outcomes compared with all other systems tested (N = 9) in the clinical validation cohort for local recurrence (5-year area under the curve (AUC) 0.81) and overall survival (5-year AUC 0.80). The increase in AUC compared with the standard of care, World Health Organization 2021 grade, was 0.11 for local recurrence (95% confidence interval 0.07 to 0.17, P &lt; 0.001). The gene expression biomarker identified meningiomas benefiting from postoperative radiotherapy (hazard ratio 0.54, 95% confidence interval 0.37 to 0.78, P = 0.0001) and suggested postoperative management could be refined for 29.8% of patients. In sum, our results identify a targeted gene expression biomarker that improves discrimination of meningioma outcomes, including prediction of postoperative radiotherapy responses