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We introduce a variant of the k-nearest neighbor classifier in which k is chosen adaptively for each query, rather than being supplied as a parameter. The choice of k depends on properties of each neighborhood, and therefore may significantly vary between different points. For example, the algorithm will use larger k for predicting the labels of points in noisy regions. We provide theory and experiments that demonstrate that the algorithm performs comparably to, and sometimes better than, k-NN with an optimal choice of k. In particular, we bound the convergence rate of our classifier in terms of a lo- cal quantity we call the “advantage”, giving results that are both more general and more accurate than the smoothness-based bounds of earlier nearest neighbor work. Our analysis uses a variant of the uniform convergence theorem of Vapnik- Chervonenkis that is for empirical estimates of conditional probabilities and may be of independent interest. 

We consider learning problems where the training set consists of two types of examples: private and public. The goal is to design a learning algorithm that satisfies differential privacy only with respect to the private examples. This setting interpolates between private learning (where private) and classical learning (where all examples are public). We study the limits of learning in this setting in terms of private and public sample complexities. We show that any hypothesis class of VC-dimension d can be agnostically learned up to an excess error of α using only (roughly) d/α public examples and d/α2 private labeled examples. This result holds even when the public examples are unlabeled. This gives a quadratic improvement over the standard d/α2 upper bound on the public sample complexity (where private examples can be ignored altogether if the public examples are labeled). Furthermore, we give a nearly matching lower bound, which we prove via a generic reduction from this setting to the one of private learning without public data. 

We present photometric and spectroscopic data on three extragalactic luminous red novae (LRNe): AT 2018bwo , AT 2021afy , and AT 2021blu . AT 2018bwo was discovered in NGC 45 (at about 6.8 Mpc) a few weeks after the outburst onset. During the monitoring period, the transient reached a peak luminosity of 10 40 erg s −1 . AT 2021afy , hosted by UGC 10043 (∼49.2 Mpc), showed a double-peaked light curve, with the two peaks reaching a similar luminosity of 2.1(±0.6)×10 41 erg s −1 . Finally, for AT 2021blu in UGC 5829 (∼8.6 Mpc), the pre-outburst phase was well-monitored by several photometric surveys, and the object showed a slow luminosity rise before the outburst. The light curve of AT 2021blu was sampled with an unprecedented cadence until the object disappeared behind the Sun, and it was then recovered at late phases. The light curve of LRN AT 2021blu shows a double peak, with a prominent early maximum reaching a luminosity of 6.5 × 10 40 erg s −1 , which is half of that of AT 2021afy . The spectra of AT 2021afy and AT 2021blu display the expected evolution for LRNe: a blue continuum dominated by prominent Balmer lines in emission during the first peak, and a redder continuum consistent with that of a K-type star with narrow absorption metal lines during the second, broad maximum. The spectra of AT 2018bwo are markedly different, with a very red continuum dominated by broad molecular features in absorption. As these spectra closely resemble those of LRNe after the second peak, AT 2018bwo was probably discovered at the very late evolutionary stages. This would explain its fast evolution and the spectral properties compatible with that of an M-type star. From the analysis of deep frames of the LRN sites years before the outburst, and considerations of the light curves, the quiescent progenitor systems of the three LRNe were likely massive, with primaries ranging from about 13 M ⊙ for AT 2018bwo , to 14 −1 +4 M ⊙ for AT 2021blu , and over 40 M ⊙ for AT 2021afy . 

ABSTRACT We present the photometric and spectroscopic evolution of supernova (SN) 2019cad during the first ∼100 d from explosion. Based on the light-curve morphology, we find that SN 2019cad resembles the double-peaked Type Ib/c SN 2005bf and the Type Ic PTF11mnb. Unlike those two objects, SN 2019cad also shows the initial peak in the redder bands. Inspection of the g-band light curve indicates the initial peak is reached in ∼8 d, while the r-band peak occurred ∼15 d post-explosion. A second and more prominent peak is reached in all bands at ∼45 d past explosion, followed by a fast decline from ∼60 d. During the first 30 d, the spectra of SN 2019cad show the typical features of a Type Ic SN, however, after 40 d, a blue continuum with prominent lines of Si ii λ6355 and C ii λ6580 is observed again. Comparing the bolometric light curve to hydrodynamical models, we find that SN 2019cad is consistent with a pre-SN mass of 11 M⊙, and an explosion energy of 3.5 × 1051 erg. The light-curve morphology can be reproduced either by a double-peaked 56Ni distribution with an external component of 0.041 M⊙, and an internal component of 0.3 M⊙ or a double-peaked 56Ni distribution plus magnetar model (P ∼ 11 ms and B ∼ 26 × 1014 G). If SN 2019cad were to suffer from significant host reddening (which cannot be ruled out), the 56Ni model would require extreme values, while the magnetar model would still be feasible. 

We present an observational study of the luminous red nova (LRN) AT 2021biy in the nearby galaxy NGC 4631. The field of the object was routinely imaged during the pre-eruptive stage by synoptic surveys, but the transient was detected only at a few epochs from ∼231 days before maximum brightness. The LRN outburst was monitored with unprecedented cadence both photometrically and spectroscopically. AT 2021biy shows a short-duration blue peak, with a bolometric luminosity of ∼1.6 × 10 41 erg s −1 , followed by the longest plateau among LRNe to date, with a duration of 210 days. A late-time hump in the light curve was also observed, possibly produced by a shell-shell collision. AT 2021biy exhibits the typical spectral evolution of LRNe. Early-time spectra are characterised by a blue continuum and prominent H emission lines. Then, the continuum becomes redder, resembling that of a K-type star with a forest of metal absorption lines during the plateau phase. Finally, late-time spectra show a very red continuum ( T BB  ≈ 2050 K) with molecular features (e.g., TiO) resembling those of M-type stars. Spectropolarimetric analysis indicates that AT 2021biy has local dust properties similar to those of V838 Mon in the Milky Way Galaxy. Inspection of archival Hubble Space Telescope data taken on 2003 August 3 reveals a ∼20 M ⊙ progenitor candidate with log ( L / L ⊙ ) = 5.0 dex and T eff  = 5900 K at solar metallicity. The above luminosity and colour match those of a luminous yellow supergiant. Most likely, this source is a close binary, with a 17–24 M ⊙ primary component. 

We present the spectroscopic and photometric study of five intermediate-luminosity red transients (ILRTs), namely AT 2010dn, AT 2012jc, AT 2013la, AT 2013lb, and AT 2018aes. They share common observational properties and belong to a family of objects similar to the prototypical ILRT SN 2008S. These events have a rise time that is less than 15 days and absolute peak magnitudes of between −11.5 and −14.5 mag. Their pseudo-bolometric light curves peak in the range 0.5–9.0 × 10 40  erg s −1 and their total radiated energies are on the order of (0.3–3) × 10 47 erg. After maximum brightness, the light curves show a monotonic decline or a plateau, resembling those of faint supernovae IIL or IIP, respectively. At late phases, the light curves flatten, roughly following the slope of the 56 Co decay. If the late-time power source is indeed radioactive decay, these transients produce 56 Ni masses on the order of 10 −4 to 10 −3   M ⊙ . The spectral energy distribution of our ILRT sample, extending from the optical to the mid-infrared (MIR) domain, reveals a clear IR excess soon after explosion and non-negligible MIR emission at very late phases. The spectra show prominent H lines in emission with a typical velocity of a few hundred km s −1 , along with Ca II features. In particular, the [Ca  II ] λ 7291,7324 doublet is visible at all times, which is a characteristic feature for this family of transients. The identified progenitor of SN 2008S, which is luminous in archival Spitzer MIR images, suggests an intermediate-mass precursor star embedded in a dusty cocoon. We propose the explosion of a super-asymptotic giant branch star forming an electron-capture supernova as a plausible explanation for these events.