The representer theorem is a cornerstone of kernel methods, which aim to estimate latent functions in reproducing kernel Hilbert spaces (RKHSs) in a nonparametric manner. Its significance lies in converting inherently infinite-dimensional optimization problems into finite-dimensional ones over dual coefficients, thereby enabling practical and computationally tractable algorithms. In this paper, we address the problem of estimating the latent triggering kernels--functions that encode the interaction structure between events--for linear multivariate Hawkes processes based on observed event sequences within an RKHS framework. We show that, under the principle of penalized least squares minimization, a novel form of representer theorem emerges: a family of transformed kernels can be defined via a system of simultaneous integral equations, and the optimal estimator of each triggering kernel is expressed as a linear combination of these transformed kernels evaluated at the data points. Remarkably, the dual coefficients are all analytically fixed to unity, obviating the need to solve a costly optimization problem to obtain the dual coefficients. This leads to a highly efficient estimator capable of handling large-scale data more effectively than conventional nonparametric approaches. Empirical evaluations on synthetic datasets reveal that the proposed method attains competitive predictive accuracy while substantially improving computational efficiency over existing state-of-the-art kernel method-based estimators.

Gabriela Cibils is on a mission - to help turn Paraguay into the Silicon Valley of South America. When she was growing up in the landlocked country, nestled between Brazil and Argentina, she says the nation wasn't super tech focused. But it was different for Ms Cibils, as her parents worked in the technology sector. And she was inspired to study in the US, where she got a degree in computing and neuroscience from the University of California, Berkeley. After graduating she spent eight years working in Silicon Valley, near San Francisco, with roles at various American start-ups.

Are there two Chilean marked gecko species, or 11? Breakthroughs, discoveries, and DIY tips sent every weekday. The Chilean marked geckos that call Chile's Atacama Desert home have proved annoyingly difficult to classify. While one might assume that different species simply look from each other, that's not always the case. Currently, Chilean marked geckos, also known as Garthia geckos, officially consist of two species-- and . However, different researchers have proposed more or even suggested that only one species exists within the genus .

Drone footage has emerged showing a blaze destroying the historic Bernaga Monastery in Italy. Founded in La Valletta Brianza in 1628, it is located about 30km (19 miles) east of Milan. More than 20 cloistered nuns were evacuated from the scene, according to Italian media reports. Could a Corrie cameo be on the cards for Daniel O'Donnell? Daniel O'Donnell said making a cameo on Coronation Street is on his bucket list.

Aftermath of RSF drone attack which killed dozens in Sudan's el-Fasher NewsFeed Aftermath of RSF drone attack which killed dozens in Sudan's el-Fasher Video shows the aftermath of drone and artillery strikes on a shelter in the besieged city of el-Fasher in Sudan's North Darfur state, which killed at least 60 people. The attack was carried out by the paramilitary Rapid Support Forces (RSF), according to a Sudanese medical advocacy group. Al Jazeera reporters follow Palestinians' return to northern Gaza Who is Nobel Peace Prize winner Maria Corina Machado?

Can Ukraine restore its pre-war borders? Why are Tomahawk missiles for Ukraine a'red line' for Russia? Is Russia testing NATO with aerial incursions in Europe? Russian drone and missile attacks across Ukraine on Saturday killed at least five people, while also cutting power to parts of the southern Odesa region, the AFP news agency reported, citing local officials. Two of the victims were killed in an attack on a church in Kostiantynivka in eastern Donetsk, AFP said.

I asked AI to plan my dream summer holiday. Here's how it turned out The gothic arches of Santa María de la Asunción crown the hill above the stone harbour where fishermen land their catches of sardines and anchovies. The church in Castro Urdiales, a small seaside town of about 30,000 people in northern Spain, is more than 700 years old. It was the perfect holiday destination for Alan Smith and his family, though he had never heard of it - until he asked ChatGPT. This week a report from the travel association ABTA found an increasing number of people were turning to AI to help with their holidays, from suggesting destinations to planning itineraries once there.

At least 60 people have been killed in a drone strike at a displacement shelter in el-Fasher, a besieged Sudanese city on the brink of collapse. The resistance committee for el-Fasher, made up of local citizens and activists, said the paramilitary Rapid Support Forces (RSF) hit Dar al-Arqam camp, located within a university, with two drone strikes and eight artillery shells. Children, women and the elderly were killed in cold blood, and many were completely burned, a statement from the group said. Eyewitnesses described scenes of panic as rescuers pulled bodies from the rubble. Hospitals already struggling under months of siege have been overwhelmed, with doctors treating the wounded on floors and in corridors.

RSF drone strike kills dozens in Sudan's war-ravaged el-Fasher: Activists A deadly drone strike at a displacement centre and university grounds, where people had been seeking refuge, in the besieged Sudanese city of el-Fasher in North Darfur state has been blamed on the Rapid Support Forces RSF), with the brutal civil war now in its third year. A local journalist told the Reuters news agency on Saturday that at least 12 people had been killed in the strike on a displacement shelter, while local activists said nearly 60 people were killed. The el-Fasher Resistance Committee said the RSF had carried out a "massacre." El-Fasher is the last major city held by the government-aligned Sudanese Armed Forces (SAF) in the vast western Darfur region, and has faced intensified attacks from the paramilitary RSF since the army recaptured the capital, Khartoum, in March this year. The RSF has been fighting SAF for control of the country since April 2023, after the two generals leading the forces fell out.

Karen Price was just 15 when she vanished in 1981 and, had it not been for a chance discovery by two builders, her body might never have been found. Because no-one was looking for her. Dubbed Little Miss Nobody, Karen had not been seen for eight years when her skeletal remains, wrapped in a carpet, were uncovered by two unsuspecting builders in Cardiff city centre on 7 December 1989. Her body, found in a shallow grave outside a basement flat on Fitzhamon Embankment, was so badly decomposed it was impossible to establish the cause of her death. Now, more than 40 years on and after the release of her killer, a new documentary has examined how police put together the jigsaw to solve the killing of a teenager known to no-one and how it involved groundbreaking methods to bring two men to justice.