Persistent homology has become an important tool for extracting geometric and topological features from data, whose multi-scale features are summarized in a persistence diagram. From a statistical perspective, however, persistence diagrams are very sensitive to perturbations in the input space. In this work, we develop a framework for constructing robust persistence diagrams from superlevel filtrations of robust density estimators constructed using reproducing kernels. Using an analogue of the influence function on the space of persistence diagrams, we establish the proposed framework to be less sensitive to outliers. The robust persistence diagrams are shown to be consistent estimators in the bottleneck distance, with the convergence rate controlled by the smoothness of the kernel — this, in turn, allows us to construct uniform confidence bands in the space of persistence diagrams. Finally, we demonstrate the superiority of the proposed approach on benchmark datasets.

We study zero-forcing (ZF) detection for multiple input/multiple output (MIMO) spatial multiplexing under transmit-correlated Rician fading for an NR× NTchannel matrix with rank-1 line-of-sight component. By using matrix transformations and multivariate statistics, our exact analysis yields the signal-to-noise ratio moment generating function (M.G.F.) as an infinite series of gamma distribution M.G.F.'s and analogous series for ZF performance measures, e.g., outage probability and ergodic capacity. However, their numerical convergence is inherently problematic with increasing Rician K-factor, NR, and NT. We circumvent this limitation as follows. First, we derive differential equations satisfied by the performance measures with a novel automated approach employing a computer-algebra tool that implements Gröbner basis computation and creative telescoping. These differential equations are then solved with the holonomic gradient method (HGM) from initial conditions computed with the infinite series. We demonstrate that HGM yields more reliable performance evaluation than by infinite series alone and more expeditious than by simulation, for realistic values of K, and even for NRand NTrelevant to large MIMO systems. We envision extending the proposed approaches for exact analysis and reliable evaluation to more general Rician fading and other transceiver methods.

For multiple-input/multiple-output (MIMO) spatial multiplexing with zero-forcing detection (ZF), signal-to-noise ratio (SNR) analysis for Rician fading involves the cumbersome noncentral-Wishart distribution (NCWD) of the transmit sample-correlation (Gramian) matrix. Anapproximationwith avirtualCWD previously yielded for the ZF SNR an approximate (virtual) Gamma distribution. However, analytical conditions qualifying the accuracy of the SNR-distribution approximation were unknown. Therefore, we have been attempting to exactly characterize ZF SNR for Rician fading. Our previous attempts succeeded only for the sole Rician-fading stream under Rician–Rayleigh fading, by writing the ZF SNR as scalar Schur complement (SC) in the Gramian. Herein, we pursue a more general matrix-SC-based analysis to characterize SNRs when several streams may undergo Rician fading. On one hand, for full-Rician fading, the SC distribution is found to be exactly a CWD if and only if a channel-mean–correlationconditionholds. Interestingly, this CWD then coincides with thevirtualCWD ensuing from theapproximation. Thus, under thecondition, the actual and virtual SNR-distributions coincide. On the other hand, for Rician–Rayleigh fading, the matrix-SC distribution is characterized in terms of the determinant of a matrix with elementary-function entries, which also yields a new characterization of the ZF SNR. Average error probability results validate our analysis vs. simulation.

For multiple-input-multiple-output (MIMO) spatial-multiplexing transmission, zero-forcing (ZF) detection is appealing because of its low complexity. Our recent MIMO ZF performance analysis for Rician-Rayleigh fading, which is relevant in heterogeneous networks, has yielded for the ZF outage probability and ergodic capacity infinite-series expressions. Because they arose from expanding the confluent hypergeometric function1F1(·, ·, σ) around 0, they do not converge numerically at realistically high Rician K-factor values. Therefore, herein, we seek to take advantage of the fact that1F1(·, ·, σ) satisfies a differential equation, i.e., it is a holonomic function. Holonomic functions can be computed by the holonomic gradient method (HGM), i.e., by numerically solving the satisfied differential equation. Thus, we first reveal that the moment generating function (m.g.f.) and probability density function (p.d.f.) of the ZF signal-to-noise ratio (SNR) are holonomic. Then, from the differential equation for1F1(·, ·, σ), we deduce those satisfied by the SNR m.g.f. and p.d.f. and demonstrate that the HGM helps compute the p.d.f. accurately at practically relevant values of K. Finally, numerical integration of the SNR p.d.f. produced by HGM yields accurate ZF outage probability and ergodic capacity results.

We analyze the performance of multiple input/multiple output (MIMO) communications systems employing spatial multiplexing and zero-forcing detection (ZF). The distribution of the ZF signal-to-noise ratio (SNR) is characterized when either the intended stream or interfering streams experience Rician fading, and when the fading may be correlated on the transmit side. Previously, exact ZF analysis based on a well-known SNR expression has been hindered by the noncentrality of the Wishart distribution involved. In addition, approximation with a central-Wishart distribution has not proved consistently accurate. In contrast, the following exact ZF study proceeds from a lesser-known SNR expression that separates the intended and interfering channel-gain vectors. By first conditioning on, and then averaging over the interference, the ZF SNR distribution for Rician-Rayleigh fading is shown to be an infinite linear combination of gamma distributions. On the other hand, for Rayleigh-Rician fading, the ZF SNR is shown to be gamma-distributed. Based on the SNR distribution, we derive new series expressions for the ZF average error probability, outage probability, and ergodic capacity. Numerical results confirm the accuracy of our new expressions, and reveal effects of interference and channel statistics on performance.